LI  BR  AR  Y 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 

GIRT    OR 


Received 

Accessions  NO._^__^J^^_     .  SJulf  fto'.._^__ 


THE 

DRAWING  GUIDE; 

A  MANUAL  OF  INSTRUCTION  IN 

INDUSTRIAL  DRAWING, 


DESIGNED  TO  ACCOMPANY  THE 


INDUSTRIAL  DRAWING  SERIES. 


WITH  AN  INTRODUCTORY  ARTICLE  ON  THE 


PRINCIPLES  AND  PRACTICE  OF  ORNAMENTAL  ART. 


BY  MARCIUS  WILLSOST, 

ATJTIIOE  or  "BCUOOL  AND  FAMILY  BEBIES  or  READEJIS,"  "MANUAL  or  OBJECT  LESSONS, 

ETC.,  ETC. 


UNIVERSITY 


HARPER  &  BROTHERS,  PUBLISHERS, 

FRANKLIN    SQUARE. 

1881. 


Entered  according  to  Act  of  Congress,  in  the  year  1873,  by 

HARPER  &  BROTHERS, 
In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


CONTENTS. 


Preface Page  v 

PART  I.   ORNAMENTAL  ART. 

I.  Introductory 13 

II.  General  Principles  of  Ornamental  Art 18 

Prop.  I.  The  Cardinal  Principle  in  Decoration,  18: — Prop.  II.  Of 
Angular  and  Winding  Forms,  19: — Prop.  III.  Of  Firm  and  Unbroken, 
and  Fine  and  Faint  Lines,  20  : — Prop.  IV.  Of  Construction  and  Deco- 
ration, 21 : — Prop.V.  Of  General  Forms,  22 : — Prop. VI.  Of  Geometrical 
Construction,  22: — Prop.  VII.  Of  Methods  of  Surface  Decoration,  23: — 
Prop.  VIII.  Of  Proportion  in  Ornamentation,  24  :— Prop.  IX.  Of  Har- 
mony and  Contrast,  25: — Prop.  X.  Of  Distribution,  Radiation,  and 
Continuity,  26 : — Prop.  XI.  Of  Conventional  Representations  of  Natu- 
ral Objects,  27. 
III.  Ornamental  Art  among  different  Nations,  and  in  different  Periods  of 

Civilization 28 

I.  Ornament  of  Savage  Tribes,  28  : — II.  Egyptian  Ornament,  29  : — 
III.  Assyrian  and  Persian  Ornament,  30  : — IV.  Greek  Ornament,  31 : 
—V.  Pompeian  Ornament,  32  :— VI.  Roman  Ornament,  32  :— VII. 
Byzantine  Ornament,  33  : — VIII.  Arabian  Ornament,  34: — IX.  Turk- 
ish Ornament,  35: — X.  Moresque  or  Moorish  Ornament,  36: — XI. 
Persian  Ornament,  37: — XII.  East  Indian  Ornament,  37: — XIII. 
Hindoo  Ornament,  38  : — XIV.  Chinese  Ornament,  38  : — XV.  Celtic 
Ornament,  39  :— XVI.  Mediaeval  or  Gothic  Ornament,  39 :—  XVII. 
Renaissance  Ornament,  41: — XVIII.  Elizabethan  Ornament,  42: — 
Modern  Ornamental  Art,  42. 


PART  II.  PRINCIPLES  AND  PRACTICE  OF  INDUSTRIAL 
DRAWING. 

DRAWING-BOOK  No.  I. 

I.  Materials  and  Directions 47 

II.  Straight  Lines  and  Plane  Surfaces 51 

Horizontal  and  Vertical  Lines,  51 : — Angles  and  Plane  Figures,  52: — 
Principles  of  Surface  Measurement,  53  :— Rules,  53-61 : — Problems, 
53 : — Diagonals,  55  : — Problems,  55  : — Two-space  Diagonals,  57 : — 
Problems,  59,  60 :— Three -space  Diagonals,  61 :  —  Problems,  62:  — 
Egyptian  Patterns,  66,  67:— Arabian,  67,  71,  72  :— Byzantine,  67,  69 : 
—  Pompeian,  67:  —  Grecian,  68-72: — East  Indian,  69: — Braided 
Work,  68  -.—Problems,  64,  66,  68. 

III.  Curved  Lines  and  Plane  Surfaces 73 

Regular  Curves,  73: — Irregular  Curves,  74: — Symmetrical  Figures, 
75  :  — Problems,  76  :  — Conventional  Leafage,  77  :  — Problems,  78 :  — 
Renaissance  Ornament,  78: — Bulb  Pattern,  79: — Problems,  79:  — 
Assyrian  and  Byzantine  Patterns,  79-81: — Problems,  81: — Quarter- 
foil,  81: — Original  Designs,  81. 

DRAWING-BOOK  No.  II. 

Cabinet  Perspective— Plane  Solids 84 

Diagonal  Cabinet  Perspective,  84  : — Elementary  Rule,  85  : — Solid 
Contents  of  a  Cube— Rule,  86 :—  Parallelepipeds— Rule,  87 :— Hatch- 
ing, 88 :  — Problems,  89  :  — Stairs,  90 :  — Cabinet  Frame-work,  91 :  — 


IV  CONTENTS. 

Problems,  92  :—  Scarfing,  92  :  —  Problems,  91 :  —Brick-work,  English 
Bond  and  Flemish  Bond,  94  : — Problems,  90  : — Divisions  of  the  Cube, 
96  :— Solid  Frets,  97 :— Timber  Framing,  97  -.—Problems,  97 :— Mould- 
ings and  Cabinet-work,  98  : — Table,  99  : — Problems,  100  : — Irregular 
Block  Forms,  100: — Problems,  102: — Pyramidal  Structures,  102:— 
Problems,l05  : — Fence  Frame-work,  1 05  : — Problems,  10G  : — Post-and- 
Kail  Fence,  106  :—  Arabian  Fret,  Solid,  107:—  Problems,  108: — Picket 
Fence,  Grecian  Frets,  Chest  with  Tray,  108, 109  : — Problems,  109  : — 
Solid  and  Hollow  Geometrical  Block  Forms,  109-111: — Bridge-work, 
112:— Cubical  Block  Patterns,  112  -.—Problems,  112. 

DPvAWING-BOOK  No.  III. 

Cabinet  Perspective — Curvilinear  Solids 114 

Cylinders,  Solid,  Hollow,  and  Divided,  114-116:—  Problems,  116  :— 
Semicircular  Arches,  117: — Problems,  118: — Braces,  Straight  and 
Curvilinear,  119  :— Quarterfoil,  120 : — Problems,  121 : — Brackets,  121 : 
—Trefoil  and  Quarterfoil,  122 -.—Problems,  123 -.—Conventional  Leaf 
Pattern,  123: — Solid  Triangle,  124: — Curvilinear  and  Quadrangular 
Solids,  125: — Architectural  Band,  126: — Problems,  126: — Rims  of 
Wheels,  126:— Beveled  Tub,  127:— Hollow  Cylinders,  128:— Irregular 
Curved  Solids,  128  : — Problems,  129  : — Curvilinear  Frame-works,  129  : 
—Problems,  131 :— Large  Wheel,  with  Spokes,  131  -.—Problem,  133  :— 
Large  Wheel,  with  Spokes  and  Double  Kim,  133  : — Crown-wheel,  135  : 
—Ratchet  Wheel,  136  :—  Windlass,  with  Spokes,  138. 

DRAWING-BOOK  No.  IV. 

Cabinet  Perspective — Miscellaneous  Applications 1 43 

I.  Different  Diagonal  Views  of  Objects 143 

II.  Ground-Plans  and  Cabinet-Plans  of  Buildings 144 

Problem,  146  : — Series  of  Platform  Structures,  147: — Problem,  149. 

III.  Cylindrical  Objects  in  Vertical  Positions 149 

I.  Ellipses  on  Diagonal  Bases 150 

II.  Ellipses  on  Rectangular  Bases 152 

Ellipses  in  Vertical  Positions,  153 : — Rule,  155 : — Problems,  15 5: — 
Hollow  Cylinders,  156  -.—Problems,  159  :— Horizontal  Wheel  with 
Spokes,  161 : — Vertical  Tub  with  Twenty-four  Uniform  Staves,  162 : 
— Problems,  163: — Crown-wheel,  with  Axis  vertical,  164: — Tub 
beveling  upward,  166: — Beveled  Octagonal  Tub,  168. 

III.  Arches  in  Diagonal  Perspective 1 69 

IV.  Semi-diagonal  Cabinet  Perspective 1 71-1 78 

V.  Shadows  in  Cabinet  Perspective 179-187 


APPENDIX. 
ISOMETRICAL  DRAWING. 

I.  Elementary  Principles 189 

II.  Figures  having  Plane  Angles 191 

III.  The  Drawing  of  Isometrical  Angles 194 

IV.  The  Isometric  Ellipse,  and  its  Applications 197 

V.  Miscellaneous  Applications 202 

VI.  Table  for  drawing  Circles  in  Isometrical  Perspective 205 

Isometric  Plates,  I.  to  VIII.  inclusive 207-221 


PREFACE. 


Ix  presenting  to  the  public  the  first  four  numbers  of  TOE 
INDUSTRIAL  DRAWING  SERIES,  a  few  words  of  explanation 
are  needed.  More  than  thirty  years  ago  the  undersigned 
prepared  a  work  on  Perspective,  Architectural,  and  Landscape 
Drawing,  for  the  use  of  an  Institution  with  which  he  was 
then  connected ;  but,  as  the  work  was  designed  for  a  local 
purpose  only,  it  has  long  been  out  of  print.  It  is  not,  there- 
fore, to  the  writer,  a  new  subject  which  he  has  now  taken 
in  hand,  but  the  elaboration  of  an  art  which,  from  boyhood, 
he  has  indulged  in  as  a  pastime,  with  constantly  enlarging 
views  of  its  importance  in  the  business  of  both  a  practical- 
ly useful  and  disciplinary  education. 

A  few  years  ago  our  attention  was  specially  called  to  the 
subject  of  Isometrical  drawing,  which  had  been  brought  for- 
ward in  England,  and  there  highly  recommended  for  the  use 
of  mechanics,  architects,  etc.,  and  for  all  purposes  in  which 
working  drawings  are  desirable.  But  the  strict  mathemat- 
ical accuracy  required  in  the  guiding  slope  lines,  which  must 
be  drawn  to  a  particular  angle,  and  for  the  drawing  of 
which  no  means  were  suggested  beyond  ordinary  pencil  rul- 
ing, placed  this  valuable  method  of  representing  objects  be- 
yond the  reach  of  all  except  the  most  accurate  draughtsmen, 
and  thus  rendered  it  almost  wholly  useless  for  all  practical 
purposes,  and  especially  for  school  uses. 

This  difficulty  in  the  ruling,  however,  we  were  enabled  to 
overcome  by  the  preparation  of  "  Isometrical  Drawing-Pa- 
per," printed  from  stone  in  fine  tinted  lines  accurately  drawn 


VI  PREFACE. 

to  the  required  angle.  We  then  proceeded  to  prepare  a 
somewhat  elaborate  work  on  Isometrical  Drawing,  in  which, 
we  have  the  assurance  to  believe,  we  were  able  to  extend 
and  simplify  the  principles  of  the  art ;  but  when  the  draw- 
ings were  all  made,  and  the  book  was  ready  for  the  press,  it 
occurred  to  us  that  a  still  more  easy  system  of  industrial 
drawing,  more  nearly  approaching  linear  perspective  in  ap- 
pearance, and  equally  practical  with  isometrical  drawing, 
might  be  invented ;  and  the  result  has  been  the  system  of 
Cabinet  Perspective,  which  is  now  offered  to  the  public  in 
the  Second,  Third,  and  Fourth  Numbers  of  the  "  Industrial 
Drawing  Series."  If  we  are  not  greatly  mistaken,  this  sys- 
tem of  Cabinet  Perspective,  which  is  so  very  simple  in  plan, 
and  so  easy  of  execution  as  to  render  its  more  valuable 
portions  capable  of  being  understood  and  practiced  by  the 
children  in  our  primary  schools,  will  give  to  the  subject  of 
industrial  drawing,  in  its  application  to  the  representation  of 
solids  of  every  variety  of  form,  a  value  hitherto  unknown. 

While  we  regard  it,  however,  as  better  for  most  industrial 
drawings,  especially  for  school  purposes,  than  Isometrical 
Perspective,  yet  the  latter  has  some  very  valuable  adapta- 
tions ;  and,  as  it  can  be  easily  applied  by  those  who  under- 
stand Cabinet  Perspective,  we  have  given  an  exposition  of 
its  principles  in  the  Appendix  to  the  present  volume. 

A  peculiarity  in  the  plan  of  the  system  now  offered  to  the 
public  consists  in  placing  the  drawings  which  are  to  be  im- 
itated, or  which  are  to  serve  as  models  for  suggesting  orig- 
inal designs,  on  paper  printed  with  fine  lines  one  eighth  of 
an  inch  apart,  and  in  furnishing  the  pupil  with  similarly 
printed  red  or  pink-lined  paper  on  which  to  make  his  draw- 
ings. These  lines  cross  each  other  at  right  angles,  vertically 
and  horizontally.  Any  draughtsman  will  see  at  a  glance 
with  what  facility  and  accuracy  a  figure  may  be  copied  from 
the  Drawing-Book  on  paper  thus  prepared  ;  how  readily  it 
may  be  enlarged  to  any  extent,  or  diminished,  in  true  pro- 


PREFACE.  Vll 

portion ;  and  how  easy  it  is,  with  the  aid  of  such  paper,  to  de- 
sign new  patterns  and  models,  and  draw  them  in  perfect  sym- 
metry in  all  their  parts.  Draughtsmen  are  often  obliged  to 
rule  paper  in  a  similar  manner,  for  their  own  use,  in  making 
intricate  patterns ;  and  it  is  perfectly  evident  that  the  vast 
variety  of  decorative  designs  which  we  find  among  the  re- 
mains of  Egyptian,  Grecian,  Roman,  Byzantine,  and  Arabian 
art,  was  formed  upon  paper,  or  papyrus,  ruled  by  pencil  in 
this  identical  manner,  although  not  on  the  scale  which  we 
have  used.  Indeed,  these  ancient  patterns  could  not  pos- 
sibly have  been  executed  with  the  accuracy  which  they  ex- 
hibit without  such  aid.  They  show  the  accurate  direction 
of  the  diagonal  and  other  oblique  lines,  which  are  so  easily 
formed  upon  such  ruling.  For  all  purposes  of  illustrating 
industrial  art,  the  two  kinds  of  ruled  drawing-paper — both 
Cabinet  and  Isometrical — will  be  found  invaluable.  Their 
varied  applications  will  be  seen  throughout  the  Industrial 
Drawing  Series. 

In  Drawing-Book  No.  I.  we  have  taken  up,  in  an  element- 
ary manner,  the  subject  of  Decorative  Design — both  on  ac- 
count of  its  being  well  adapted  to  elementary  practice  in 
drawing,  and  because  of  its  importance  in  nearly  all  depart- 
ments of  industrial  art. 

In  our  drawing-lessons  under  this  head,  we  have  aimed, 
in  the  first  place,  to  furnish  a  variety  of  such  copies  as  are 
most  suitable  for  elementary  exercises  in  training  the  hand 
and  the  eye,  while  at  the  same  time  they  shall  be  adapted  to 
cultivate  a  correct  taste  for  that  which  combines  harmony  of 
design  with  grace  and  beauty  of  form.  Hence,  instead  of 
thinking  it  desirable  that  we  should  originate  all  of  our 
figures  for  the  drawing  exercises,  we  have  selected  them,  in 
great  part,  from  the  best  examples  of  the  decorative  art  of 
all  ages,  being  parts,  or  wholes,  of  patterns  which  have  stood 
the  test  of  time,  the  only  true  standard  of  taste.  By  this 
course  we  are  not  only  able  to  give  a  very  great  variety  of 


Vlll  PREFACE. 

excellent  patterns  for  imitation,  and  for  suggestion  in  de- 
signing, but  we  are  also  enabled  to  impart  to  the  pupil  some 
general  knowledge  of  those  principles  of  form  and  propor- 
tion which  govern  all  true  art  decoration  ;  and  in  the  intro- 
ductory articles  we  have  given  brief  sketches  of  the  growth 
and  development  of  these  principles  in  different  nations  and 
in  different  periods  of  civilization.  Should  the  Series  be  car- 
ried so  far  as  we  now  anticipate,  we  hope,  in  higher  numbers, 
to  greatly  enlarge  upon  the  designs  here  given ;  to  show  the 
application  of  industrial  drawing  to  various  specific  forms  of 
industry ;  and  also  to  illustrate  the  Harmonies  of  Color,  as 
applied  to  decorative  art. 

But  we  would,  furthermore,  call  special  attention  to  the 
new  method  of  representing  objects,  called  CABINET  PER- 
SPECTIVE, as  illustrated  in  Drawing-Books  Numbers  II.,  III., 
and  IV.,  and  embracing  both  plane -and  curvilinear  solids  in 
almost  every  variety  of  form  and  position.  This  kind  of  per- 
spective, when  carried  out  by  the  use  of  the  ruled  drawing- 
paper,  enables  us  to  construct  all  kinds  of  working  drawings 
for  artisans — drawings  which,  instead  of  giving  a  geometrical 
representation  of  but  one  side  of  a  rectilinear  object,  present 
in  one  view  three  sides,  at  the  same  time  avoiding  the  appear- 
ance of  distortion,  and  giving,  with  perfect  accuracy,  the  same 
as  Isometrical  Perspective,  the  dimensions  of  the  objects  rep- 
resented, according  to  whatever  scale  the  draughtsman  may 
adopt.  Moreover,  the  principles  of  the  system  are  so  simple 
that  a  child  can  understand  them ;  while  any  one  who  can 
draw  straight  lines  by  the  aid  of  a  ruler,  and  curved  lines 
by  the  aid  of  a  pair  of  compasses,  can  apply  them. 

As  indicating  something  of  the  scope  of  the  system,  as  ap- 
plied to  solids,  we  have  represented,  under  this  head,  within 
the  narrow  limits  which  we  have  assigned  to  ourselves,  such 
objects  as  cabinet  frame-works  of  various  forms;  tables;  cu- 
bical, hexagonal,  octagonal,  and  other  blocks,  either  entire, 
or  variously  cut  and  divided;  crosses;  star  figures;  boxes; 


PREFACE,  x 

English  bond  and  Flemish  bond — forms  of  brick-laying  ;  pil- 
lars and  their  mouldings  ;  pyramids,  obelisks,  etc. ;  post  and 
board,  post  and  rail,  and  picket  fences ;  various  forms  of  the 
solid  Grecian  fret,  and  other  architectural  ornaments ;  frame- 
work of  bridges ;  cylinders,  solid  or  hollow,  entire,  or  various- 
ly cut  and  divided,  and  in  both  horizontal  and  vertical  po- 
sitions ;  arches,  both  pointed  and  semicircular ;  braces  and 
brackets,  both  plain  and  curvilinear;  solid  quarterfoils  and 
trefoils ;  wheels,  in  sections,  and  entire — with  crown-wheel, 
ratchet-wheel,  chain-pulley  wheel,  etc. ;  windlass ;  vertical 
and  beveled  tubs,  both  circular  and  octagonal ;  ground-plans 
and  elevations  of  buildings  ;  tenon  and  mortise  work  ;  scarf- 
jointing  of  timbers;  stairways;  platforms;  ellipses;  rings,  etc., 
etc.,  and  all  drawn  to  definite  dimensions,  while  the  measure- 
ments are  indicated  by  the  drawing-paper  itself.  By  this 
system  the  study  of  drawing,  in  its  application  to  the  indus- 
trial arts,  is  rendered  one  of  the  exact  sciences,  wholly  me- 
chanical in  execution,  and  as  accurate  in  its  delineations  as 
geometry  itself.  We  have  here  presented  only  an  elementary 
exposition  of  the  system,  designed  for  school  purposes ;  but 
the  system  itself  is  so  simple,  that,  with  the  helps  here  given, 
the  intelligent  teacher  and  pupil  will  find  little  difficulty  in 
carrying  out  the  application  of  its  principles  to  any  extent 
which  may  be  desired. 

For  several  of  the  rules  and  principles  of  Ornamental  Art, 
and  also  for  many  of  the  designs  in  Drawing-Book  Xo.  I., 
we  are  indebted  to  the  "  Grammar  of  Ornament"  by  Owen 
Jones.  It  may,  perhaps,  be  thought  that  it  was  not  especial- 
ly desirable  to  preface  an  Elementary  Drawing  Series  with 
a  statement  of  the  general  principles  of  Art  Decoration,  and 
an  account  of  the  Leading  Schools  or  Periods  of  Art,  for  the 
reason  that  such  information  will  seldom  be  appreciated  by 
beginners  in  drawing.  But  to  teachers  at  least — and  not 
merely  teachers  of  drawing — we  may  hope  that  these  intro- 
ductory pages  will  be  of  some  value;  and  if  they  shall  serve 

A2 


X  PREFACE. 

merely  to  enlarge  the  ideas  of  both  teachers  and  pupils  as  to 
the  magnitude  and  importance  of  the  subject  of  art  repre- 
sentation, they  will  thereby  have  done  a  good  service  to  the 
cause  of  education. 

We  would  take  this  occasion  to  impress  upon  educators, 
and  those  who  have  the  management  of  our  Public  Schools, 
the  extreme  desirability  that  all  the  school  instruction  in  ele- 
mentary industrial  drawing  shall  be  given  by  the  ordinary 
teachers ;  and  that  professional  drawing-masters  shall  be  em- 
ployed, if  at  all,  only  in  the  training  of  the  teachers  them- 
selves—in a  general  superintendence  of  the  whole  subject  of 
art  instruction  in  all  the  schools  of  a  city,  or  county,  or  even 
larger  district — or  in  giving  instruction  to  advanced  stu- 
dents in  the  higher  Schools  of  Design.  The  teachers  in  our 
Public  Schools  are  competent  to  give  all  the  instruction  re- 
quired by  their  classes  in  industrial  drawing ;  and  care  should 
be  taken  that  pupils  do  not  get  the  idea  that  they  are  re- 
quired to  do  something  which  their  teachers  themselves  can 
not  do. 

MARCIUS  WILLSON. 

VINELAND,  N.  J.,  June  5,  1873. 


PART  I. 

PRINCIPLES  AND  PRACTICE 


OP 


ORNAMENTAL  ART. 


L  ^StomifiK 


INTRODUCTORY. 


WE  desire  to  offer  to  the  public  a  few  introductory  re* 
marks  on  Ornamental  Art,  a  subject  which  we  have  en- 
deavored to  illustrate,  in  a  very  elementary  manner,  in  the 
first  book  of  our  Industrial  Drawing  Series. 

We  are  aware  that  those  who  have  given  the  subject  but 
little  attention  entertain  very  erroneous  ideas  of  the  im- 
portance and  value  of  a  knowledge  of  the  principles  and 
practice  of  decoration,  as  applied  to  the  products  of  human 
industry.  A  very  little  reflection,  however,  must  convince 
the  most  utilitarian,  that,  in  an  advanced  stage  of  society, 
decoration  enters  so  fully  into  all  works  of  art  as  to  consti- 
tute, in  perhaps  a  majority  of  cases,  the  greater  part  of 
their  market  value.  We  see  the  principle  illustrated  in  the 
importance  that  is  attached  to  surface  ornamentation  in  the 
manufacture  of  carpets,  and  oil-cloths,  and  matting,  and 
wall-paper,  and  curtains;  in  printed  cloths,  and  other  arti- 
cles designed  for  dress  ;  in  crochet  and  tapestry  work  ;  in 
the  elegant  forms  required  for  vases,  and  all  crockery  and 
earthenware  ;  alike  in  the  fine  sculpture  of  the  most  delicate 
ornaments  and  the  chiseling  of  stone  for  public  and  private 
dwellings;  in  all  mouldings  of  wood,  and  iron,  and  other 
ornamental  work  in  architecture  ;  and  it  is  found  to  enter 
into  all  plans  and  patterns  of  utensils  and  tools,  and  into 
all  objects  of  art  which  may  be  deemed  capable  of  improve- 
ment by  giving  to  them  increased  beauty  of  form  and  pro- 
portion. Indeed,  all  the  vast  variety  of  form  and  color 
which  we  observe  in  the  works  of  man,  beyond  the  require- 
ments of  the  most  barren  utility,  is,  simply,  ornamentation. 
Beginning  with  the  savage,  with  whom  ornament  precedes 
dress,  it  has  been  the  study  of  man  in  all  ages  not  only  to 
make  art  beautiful,  but  to  improve  upon  nature  also.  The 


14  INDUSTRIAL   DRAWING. 

subject  is  thus  seen  to  embrace  all  that,  in  industrial  art, 
marks  the  advance  of  civilization ;  and  decoration  may  be 
taken  as  a  true  exponent,  in  every  stage  of  its  development, 
of  the  progress  of  society ;  for  the  comforts  and  the  elegan- 
cies of  life  are  ever  found  to  grow  together. 

Inasmuch,  therefore,  as  ornamentation  enters  so  largely 
into  the  daily  life  of  civilized  society  as  to  be  every  where 
recognized,  studied,  admired,  and  practiced,  it  would  seem 
not  only  appropriate,  but  very  desirable,  that  its  elementary 
principles,  at  least,  should  find  a  place  at  the  beginning  of 
every  system  of  public  instruction — and,  where  they  prop- 
erly belong,  in  the  study  and  practice  of  Industrial  Drawing. 

England  is  so  decidedly  a  manufacturing  country,  that 
art  education  has  there  long  been  deemed  a  national  neces- 
sity :  and  it  is  not  only  thought  important  that  the  manufac- 
turer should  understand  the  laws  of  beauty,  and  the  princi- 
ples of  design,  in  order  that  his  products  may  command  a 
ready  market,  but  that  the  artisan  also — the  mere  workman 
in  art — shall  possess  something  of  the  skill  which  comes 
from  educated  taste.  More  than  thirty  years  ago  a  British 
Association  for  the  Advancement  of  Art,  composed  of  the 
chief  nobility,  capitalists,  bankers,  merchants,  and  manufac- 
turers of  the  kingdom,  sent  out  the  declaration  and  appeal, 
that,  without  a  pre-eminence  in  the  arts  of  design,  British 
manufacturers  could  not  retain,  and  must  eventually  lose, 
their  superiority  in  foreign  markets.  But  the  English  gov- 
ernment remained,  for  years,  deaf  to  the  warning ;  and  at 
the  great  Exhibition  of  the  Industry  of  all  Nations,  held  in 
London  in  1851,  England  found  herself  almost  at  the  bot- 
tom of  the  list  in  respect  to  excellence  of  design  in  her 
art  manufactures — only  the  United  States,  among  the  great 
nations,  being  below  her.  This  discovery  aroused  the  En- 
glish government  to  a  realizing  sense  of  the  vast  importance 
of  the  highest  and  most  widely  diffused  art  education  for  a 
manufacturing  people ;  and  the  result  was  the  speedy  estab- 
lishment of  an  Educational  Department  of  Science  and  Art, 
from  which  Schools  of  Design  have  radiated  all  over  the 
country.  In  these  and  other  schools,  even  ten  years  ago, 
two  thousand  students  were  in  training  as  future  teachers 


ORNAMENTAL   AKT.  15 

of  art,  and  fifteen  thousand  pupils  were  receiving  an  art 
education ;  while  in  the  parish  and  public  schools  more  than 
fifty  thousand  children  of  the  laboring  and  poorer  classes 
were  receiving  more  or  less  instruction  in  elementary  draw- 
ing. In  the  higher  art  schools  the  pupils  are  taught  not 
only  the  practice,  but  the  principles  also,  of  ornamental  de- 
sign ;  they  are  shown  how  all  assemblages  of  ornamental 
forms  are  arranged  in  geometrical  proportions :  how  curves 
must  flow,  the  one  into  the  other,  without  break  or  interrup- 
tion ;  and  they  are  taught  to  analyze  and  interpret  the  char- 
acteristic ideas  of  various  styles  and  schools  of  art,  such  as 
we  have  given  a  brief  synopsis  of  under  the  heading  of 
"  Ornamental  Art  among  Different  Nations,  and  in  Different 
Periods  of  Civilization."  The  wisdom  of  England's  course 
was  very  apparent  at  the  Paris  Exhibition  of  1867,  when 
it  was  seen  that  England  had  risen,  in  a  period  of  six- 
teen years,  from  a  position  among  the  lowest,  to  one  fore- 
most among  the  nations  in  art  manufactures — showing  the 
effects  of  the  art  education  which  she  had  so  sedulously  fos- 
tered. As  humiliating  as  it  is  to  our  national  pride,  truth 
compels  us  to  add,  in  the  language  of  another — "  The  United 
States  still  held  her  place  at  the  foot  of  the  column."  In 
England,  in  1870,  besides  the  attention  given  to  drawing  in 
the  public  schools  and  in  evening  classes,  there  were  more 
than  twenty  thousand  students  in  the  art  schools,  and  more 
than  thirty  thousand  in  the  schools  of  industrial  science ; 
and  it  is  reported  that,  in  the  two  following  years,  the  num- 
bers in  both  were  doubled. 

A  notable  illustration  of  the  commercial  value  of  the 
beautiful  in  art  is  afforded  in  the  colossal  growth  of  the 
earthenware  trade  in  England,  which  started  into  sudden 
notoriety  when  the  young  sculptor,  Flaxman,  was  employed 
to  model,  from  fine  specimens  of  antique  sculpture,  those 
beautiful  urns,  vases,  goblets,  and  other  articles  for  table 
service  and  other  domestic  uses,  long  known  as  the  Wedge- 
wood  ware.  The  clay  pits  of  Staffordshire  were  turned  into 
gold  mines,  and  made  a  source  of  national  wealth,  when  the 
proprietors  employed  good  artists  to  draw  designs  and  se- 
lect antique  models  for  their  workmen;  and  it  has  been 


16  INDUSTRIAL   DBAWIXG. 

stated  by  competent  judges  that,  through  the  establishment 
of  Art  Museums  and  Schools  of  Design,  and  the  influences 
exerted  thereby,  combined  with  popular  instruction  in  in- 
dustrial drawing,  England  has  added  fifty  per  cent,  to  the 
value  of  her  manufactured  products  during  the  past  twenty- 
five  years.  And,  turning  to  the  Continent,  we  find  it  is  the 
art  instruction  imparted  in  the  schools  and  in  the  manufac- 
tories of  France,  showing  how  colors  are  distributed,  bal- 
anced, and  harmonized,  both  in  nature  and  in  art,  that  has 
given  to  the  silk  fabrics  of  Lyons,  the  Gobelin  tapestry,  and 
to  other  national  products,  their  world-wide  renown  for  har- 
mony and  beauty.  In  France,  education  in  science  and  art 
is  now  placed  by  law  in  the  same  rank  as  classical  education. 
In  our  own  country  public  attention  is  now  being  turned, 
in  a  very  marked  manner,  to  the  subject  of  art  education : 
and  in  Massachusetts,  after  the  subject  had  been  agitated 
by  the  leading  manufacturers  and  merchants,  laws  have 
been  passed  securing  to  pupils  instruction  in  elementary 
drawing  in  every  public  school  in  the  state ;  making  u  in- 
dustrial or  mechanical  drawing"  free  to  persons  over  fif- 
teen years  of  age  either  in  day  or  evening  classes,  in  cities 
and  towns  that  have  a  population  above  ten  thousand ;  and 
a  State  Director  of  Art  Education  has  been  appointed  to 
supervise  the  system;  but,  generally,  throughout  our  schools, 
what  little  imperfect  instruction  in  art  has  been  given  has 
thus  far  been  confined,  mostly,  to  the  mere  copying  of  pic- 
tures— and,  where  it  has  gone  beyond  that,  to  the  education 
of  artists  rather  than  of  artisans.  It  is  seldom  addressed, 
as  it  should  be,  to  the  principles  and  practice  of  ornamental 
design ;  to  the  harmonies  of  color,  form,  and  proportion ; 
and  to  such  representations  of  objects  as  are  most  needed 
by  workingmen  in  the  arts.  This  kind  of  art  knowledge  and 
practice  would  not  only  be  of  interest,  but  of  utility  to  all ; 
and  the  mechanic  who  could  make  the  best  use  of  it  in  his 
line  of  business  would  ever  have  a  decided  advantage  over 

O 

all  competitors.  An  incident  bearing  upon  this  point  is  re- 
lated by  the  State  Director  of  Art  Education  for  Massa- 
chusetts, to  the  effect  that,  "  some  years  ago  a  class  of  thir- 
teen young  men  spent  all  their  leisure  time  in  studying 


OEXAMENTAL  AKT.  17 

drawing,  and  that  now,  at  the  time  of  writing,  every  one  of 
them  holds  some  important  position,  either  as  manufacturer 
or  designer."  And  if  we  would  build  up  our  manufactories 
on  a  broad  scale,  so  as  to  bring  their  products  into  success- 
ful competition  with  those  of  England  and  France,  we  must 
not  rely  upon  a  few  imported  draughtsmen  and  designers, 
and  vainly  hope  that  uneducated  artisans  will  work  out 
foreign  patterns  with  taste  and  beauty;  but  we  must  lay 
the  foundations  of  art  superiority  broad  and  deep  in  the  art 
education  of  all  mechanics,  and  in  the  educated  tastes  of  the 
people.  Then  draughtsmen  and  designers  will  spring  up 
wherever  needed ;  and  the  workmen  in  our  shops  and  manu- 
factories, understanding  the  principles  of  their  several  trades 
and  professions,  will  be  all  the  more  skilled  in  the  practice 
of  them.  And  what  we  need  for  this  is  not  merely  a  few 
Schools  of  Design,  and  Art  Museums,  valuable  as  these  may 
be,  but  the  introduction  of  the  principles  of  design,  and  the 
practice  of  art  representation,  into  the  education  of  the 
people  at  large. 

But  here  the  practical  question  is  suggested :  How  shall 
we  introduce  Industrial  Drawing  into  our  schools,  so  that 
all  our  youth  may  profit  by  it,  when  so  many  other  impor- 
tant studies  are  crowding  for  admission,  and  our  teachers 
have  already  quite  as  much  as  they  can  attend  to?  We 
reply,  Alternate  it  with  the  writing -lessons;  and  experi- 
ence fully  proves  that  better  penmanship  will  be  attained 
thereby,  while  the  drawing,  and  the  knowledge  which  it 
introduces,  will  be  a  positive  gain,  without  any  attendant 
loss.  Long  ago,  said  that  veteran  educator,  Horace  Mann, 
"  I  believe  a  child  will  learn  both  to  draw  and  write  sooner, 
and  with  more  ease,  than  he  will  learn  writing  alone." 

In  conclusion,  wre  commend  this  wrhole  subject  of  Indus- 
trial Art  Education  as  worthy  the  earnest  consideration, 
not  only  of  all  educators,  but  also  of  all  mechanics  and  ar- 
tisans, and  of  all  who  appreciate  the  vast  proportions  which 
our  manufacturing  interests  are  destined  to  assume  in  the 
not  far  distant  future. 


II. 

GENERAL  PRINCIPLES  OF  ORNAMENTAL  ART. 

THERE  are  two  kinds  of  beauty  in  Ornamental  Art :  the 
one  is  the  beauty  of  design  and  execution,  arising  from  the 
exhibition  of  skill  on  the  part  of  the  designer  and  artisan ; 
the  other  is  the  beauty  of  character,  which  arises  from  the 
expression  of  thought  or  soul  in  the  object  itself.  The 
beauty  of  the  former  is  fully  realized  only  by  those  who 
are  proficients  in  the  art,  and  ceases  to  be  felt  when  the  art 
has  made  a  farther  progress.  The  beauty  of  the  latter,  in- 
asmuch as  it  appeals  to  the  sensibilities  of  all,  is  universally 
felt,  although  in  a  different  degree  by  different  individuals, 
and  is  by  far  the  most  lasting  ;  and  the  former  should  ever 
be  subordinate  to  it.  The  difference  between  the  two  kinds 
of  beauty  is  best  illustrated  in  architecture,  of  which  orna- 
ment is  the  very  soul  and  spirit.  All  that  utility  requires 
in  the  structure,  skill  may  accomplish  by  the  aid  of  mere 
rule  and  compass ;  but  the  ornamentation  shows  how  far  the 
architect  was,  at  the  same  time,  an  artist. 

PROPOSITION  I. — A  CARDINAL  PRINCIPLE. 
All  decoration  should  exhibit  a  fitness  or  propriety  of 
things,  just  proportions,  and  harmony  of  design. 

All  ornaments  should  harmonize  in  expression  with  the 
expression  designed  to  be  given  to  the  objects  to  which 
they  are  affixed.  Thus  there  are  art  objects  of  convenience 
and  use,  of  sublimity,  of  splendor,  of  magnificence,  of  gayety, 
of  delicacy,  of  melancholy,  etc. ;  and  the  ornaments  affixed 
to  each  should  fully  harmonize  with  its  character.  Any 
fabric  to  be  ornamented  should,  in  the  first  place,  be  suited 
to  its  proposed  uses ;  and  then,  in  strict  keeping  with  the 
main  design,  must  be  the  decoration  which  adorns  its  sur- 
face. Hence,  to  cover  an  oil-cloth,  or  a  chair  cushion,  with 


ORNAMENTAL   AET.  19 

drawings  of  cubical  blocks  set  on  edge,  as  we  have  seen,  is 
an  outrage  upon  the  uses  to  which  either  is  to  be  put ;  and 
alike  improper  is  it  to  load  a  carpet,  designed  for  the  tread 
of  feet,  with  vases  filled  with  fruits,  and  to  cover  it  thick 
with  garlands  of  flowers.  It  is  only  in  the  richest  velvet 
carpets,  elastic  to  the  tread,  and  where  the  flowers  are  par- 
tially lost  in  the  profusion  of  herbage,  that  such  excessive 
adornment  may  be  deemed  not  inappropriate.  As  is  well 
known,  the  Greek  orders  of  architecture  have  manifest  dif- 
ferences of  character  or  expression.  Thus  the  heavy  Tus- 
can is  distinguished  by  its  severity;  the  manly  Doric  by  its 
simplicity,  purity,  and  grandeur ;  the  Ionic  by  its  grace  and 
elegance;  the  Corinthian  by  its  lightness,  delicacy,  and 
gayety;  and  the  Composite  by  its  profusion  and  luxury; 
while  the  ornaments  of  the  several  orders  fully  harmonize 
with  them  in  expression.  Thus  every  product  of  art  has 
some  character  of  its  own,  and  good  taste  demands  that 
there  shall  be  a  correspondence  in  the  decoration  given  to 
it.  A  degree  of  ornamentation  that  would  be  becoming  in 
one  object,  would  be  insipid  or  mean  in  another; — as  what 
would  be  in  good  taste,  and  beautiful,  in  the  robes  of  a 
queen,  would  be  inappropriate  in  the  dress  of  a  plain  lady, 
and  tawdry  in  that  of  a  peasant  girl.  And  although  wreaths 
of  flowers  may  alike  deck  the  tomb  and  adorn  the  festive 
hall,  yet  the  variety  and  profusion  suited  to  the  latter  would 
not  comport  with  the  subdued  feeling  which  is  in  unison 
with  the  former;  and  the  true  artist  will  at  once  discern 
the  difference.  The  vase  upon  a  tomb  will  not  bear  the  va- 
riety of  contour  that  may  be  given  to  a  goblet ;  nor  should 
the  latter  have  the  uniformity  of  moulding  characteristic  of 
a  funeral  urn. 

PROPOSITION  II. — OF  ANGULAR  AND  WINDING  FORMS. 

Angular  forms  denote  harshness,  maturity,  strength,  and 
vigor.     Winding  forms,  on  the  contrary,  are  expressive  of 
infancy,  weakness,  tenderness,  and  delicacy,  as  also  of  ease,   . 
grace,  beauty,  luxury,  and  freedom  from  force  and  restraint. 

As  in  all  objects  of  taste  the  lightest  forms  consistent 
with  the  required  strength  are  considered  the  most  beauti- 


20  INDUSTRIAL    DRAWING. 

fill,  so  in  all  articles  in  which  much  strength  is  required 
angular  forms  are  generally  adopted,  because  they  require 
a  less  amount  of  material  than  curvilinear  forms.     Hence 
angular  forms — as  of  squares,  lozenges,  etc. — are  not  only 
best  suited  to  such  articles  of  furniture  as  chairs,  tables, 
desks,  stands,  etc.,  but  also  to  oil-cloths,  matting,  plain  car- 
pets, etc.,  because  we  associate  with  these  latter  articles 
much  tread  of  feet  and  daily  use  ;  and  yet  it  is  equally  ap- 
parent that  these  angular  forms  would  not  be  appropriate 
for  carpets  of  luxurious  ease,  for  flowing  robes,  curtains, 
etc.     In  architecture  we  expect  direct  and  angular  lines, 
because  they  give  the  impression  of  stability  and  strength ; 
and  architectural  ornaments  are  beautiful  only  as  they  are 
in  harmony  with  the  general  character  of  the  structure  to 
which   they  are   affixed.     An   angular  vase,  designed  for 
holding  flowers,  would  be  exceedingly  inappropriate ;  while, 
on  the  contrary,  to  make  the  sides  of  a  house,  or  of  a  pyra- 
mid, curvilinear,  would  none  the  less  violate  our  ideas  of 
[fitness  and  propriety.     The  weeping  willow,  as  it  is  appro- 
priately named,  is  adapted  to  mournful  occasions,  because 
|  it  bends  and  droops  like  one  in  affliction ;  while  the  sturdy 
j.oak,  on  the  contrary,  of  angular  outlines,  is  representative 
Vof  firmness  and  strength.     It  may  break,  but  can  not  bend. 

PROPOSITION  III. — OF  FIRM  AND  UNBROKEN,  AND  FINE  AND 
FAINT  LINES. 

Firm  and  unbroken  lines  are  expressive  of  strength  and 
boldness,  with  some  degree  of  harshness. 

Fine  and  faint  lines  are  indicative  of  smoothness,  fine- 
ness, delicacy,  and  ease. 

When  the  forms  of  objects  are  used  to  ornament  articles 
of  taste  or  utility,  they  should  be  drawn  in  keeping  with 
the  character  of  the  objects  themselves.  Thus  the  visual 
line  of  a  column,  or  of  a  pyramid,  should  be  bold  and  un- 
broken, unless  modified  by  distance*  of  view;  while  the 
winding  outlines  of  the  tendrils  of  a  vine,  of  a  wreath,  of  a 
festoon,  should  be  exceedingly  \delicate,  as  we  say — our  very 
language  conforming  to  our  ideas  of  the  fitness  of  things. 
But  see  Proposition  XI. 


ORNAMENTAL    ART.  21 

PROPOSITION  IV. — OF  CONSTRUCTION  AND  DECORATION. 

Construction  should  be  decorated ;  bat  decoration  should 
never  be,  purposely  constructed. 

In  the  weaving  of  lace,  muslin,  and  other  fabrics  of  one 
color,  in  a  variety  of  suitable  patterns,  and  in  the  similar 
braiding  of  mats,  baskets,  etc.,  the  construction  itself  is,  ap- 
propriately, decorated.  So  may  any  construction  —  as  a 
building,  a  robe,  an  article  of  furniture,  etc. — be  decorated  in 
the  making  of  it ;  but  to  construct  or  plan  a  decoration  with- 
out regard  to  the  application  or  use  that  is  to  be  made  of 
it,  and  as  if  it  might  serve  a  variety  of  purposes,  is  a  viola- 
tion of  the  principles  of  true  art.  It  is,  therefore,  the  cor- 
rect principle,  to  make  the  construction  itself  ornamental, 
rather  than  to  depend  upon  applied  ornament.  Hence  the 
veneering  of  the  fronts  of  brick  or  wooden  buildings  with 
marble,  or  articles  of  wooden  furniture  with  thin  layers  of 
richer  wood,  is  a  sham  that  gives  us  a  feeling  of  disappoint- 
ment when  the  cheat  is  known.  So  the  painting  or  grain- 
ing of  wood  is  far  less  satisfactory,  as  a  decorative  agent, 
than  the  bringing  out  and  preservation  of  the  natural  grain 
by  a  suitable  varnish.  Artistic  arrangements  of  American 
woods  properly  prepared  would  furnish  a  wonderful  variety, 
in  pattern  and  coloring,  for  decorative  purposes,  and  in  far 
better  taste  than  most  of  the  surface  decoration  that  is  pur- 
posely constructed. 

Every  object  of  art  production  is  supposed  to  be  con- 
structed with  some  definite  aim,  and  to  be  designed  to  sub- 
serve some  purpose  of  utility ;  or,  if  it  be  merely  ornament- 
al, it  is  still  designed  to  aid  in  giving  the  true  and  proper 
expression  to  that  object  to  which  it  is  affixed.  In  either 
case,  the  style,  character,  and  expression  of  the  ornamental 
are  to  be  considered  as  the  accessories,  and  to  be  governed 
wholly  by  the  character  of  the  object  of  which  they  are 
the  appendages.  A  carpet,  a  dress,  a  curtain,  or  a  chair, 
etc.,  should  be  ornamented  with  reference  to  the  circum- 
stances and  occasions  of  its  uses ;  and,  evidently,  it  must 
vary  in  decoration  according  as  it  may  be  designed  for  a 
cottage  or  for  a  palace.  So  mere  ornaments,  as  rings,  brace' 


22  INDUSTRIAL   DRAWING. 

lets,  brooches,  etc.,  should  be  adapted  to  the  character,  per- 
sonal appearance,  and  position  in  society  of  the  wearer; 
for,  not  all  beautiful  things  are  becoming  to  all  places,  or 
to  all  persons.  The  proprieties  of  life  have  a  very  wide 
range  of  application.  See  Proverbs  xi.,  22. 

PROPOSITION  V. — OF  GENERAL  FORMS. 

True  beauty  of  form  is  produced  by  lines  growing  out  of 
one  another  in  gradual  undulations,  and  supported  by  one 
another.  There  are  no  excrescences  ;  and  nothing  could  be 
removed  and  leave  the  design  equally  good  or  better. 

These  principles  are  best  illustrated  in  the  several  orders 
of  Grecian  architecture,  from  no  one  of  which  could  any 
portion  be  taken  away  without  leaving  the  general  form 
defective ;  and  certainly  no  part  could  be  enlarged  without 
giving  to  it  the  appearance  of  an  unseemly  excrescence. 

PROPOSITION  VI. — OF  GEOMETRICAL  CONSTRUCTION. 

All  surface  ornamentation  should  be  based  upon  a  geo- 
metrical construction. 

Whatever  the  pattern  of  the  ornament,  it  should  be 
such  that  it  can  be  traced  back  to  a  geometrical  basis ;  and 
no  ornament  can  be  properly  designed  without  such  aid  as  a 
groundwork.  Especially  is  this  the  case  in  woven  fabrics, 
which  are  necessarily  constructed  on  a  geometrical  plan. 

As  in  the  infancy  of  art  uniformity  of  design  was  most 
valued,  as  evincing  the  skill  of  the  artist;  and  as  what  chil- 
dren most  admire,  and,  in  their  little  attempts  at  art,  first 
try  to  execute,  is  uniformity  and  regularity,  so  elementary 
drawing  should  begin  with  those  simple  geometrical  pat- 
terns which  are  the  groundwork  of  all  artistic  ornamentation. 

Patterns  in  which  the  geometrical  arrangement  is  at 
once  apparent,  owing  to  the  uniformity  or  regularity  of 
the  details,  owe  the  first  impression  of  beauty  which  they 
give  us  to  their  expression  of  design  on  the  part  of  the  art- 
ist; and  the  more  intricate  the  pattern,  and  the  greater 
the  number  of  its  parts,  while  it  still  preserves  its  uniform- 
ity, the  higher,  in  the  estimation  of  educated  taste,  is  its  de- 
gree of  beauty ;  only  the  number  of  parts  must  not  be  so 


ORNAMENTAL    ART.  23 

great  as  to  produce  confusion,  and  thus  obscure  the  expres- 
sion of  design.  Where,  however,  a  confused  intricacy  of 
detail  at  first  seems  to  prevail,  nothing  is  more  delightful 
than  to  find  order  gradually  emerging  out  of  chaos,  and  a 
consistent  plan  pervading  the  whole.  When  there  is  add- 
ed to  a  beautiful  design  intricacy  and  variety  of  detail 
amid  uniformity,  there  is  only  needed  elegance  and  embel- 
lishment in  the  workmanship  to  constitute  the  highest  de- 
gree of  ornamental  art. 

PROPOSITION  VII. — OF  METHODS  OF  SURFACE  DECORATION. 

The  general  forms  of  the  desired  ornamentation  having 
been  first  drawn  on  some  geometrical  basis,  consistent  with 
the  character  of  the  object  to  be  ornamented,  these  forms 
should  then  be  subdivided  and  ornamented  by  general  lines  ; 
the  intermediate  spaces  may  then  be  filed  in ;  and  the  sub- 
division may  be  continued  to  any  extent  required,  and  until 
the  details  can  be  appreciated  only  by  close  inspection. 

This  method  of  designing  is  adapted  no  less  to  the  some- 
times elaborate  patterns  of  embroidered  robes  and  tapestry 
work,  than  to  the  more  obvious  geometrical  arrangements 
of  squares,  and  parallelograms,  and  lozenges,  and  circles, 
that  are  often  seen  in  oil-cloths  and  carpeting.  The  great 
secret  of  success,  even  in  the  most  complicated  ornamenta- 
tion, is  the  production  of  a  broad  general  effect  by  the  rep- 
etition of  a  few  simple  elements.  u  Variety  should  rather 
be  sought  in  the  arrangement  of  the  several  portions  of  a 
design,  than  in  the  multiplicity  of  varied  forms." 

In  the  wall  or  floor  ornamentation  of  dwellings,  an  im- 
portant principle  to  be  observed  is  the  use  of  modest  tints 
as  a  back-ground,  against  which  the  furniture  can  be  dis- 
played to  advantage,  and  a  due  subordination  to  the  uses 
to  which  the  room  is  to  be  applied — as,  for  example,  wheth- 
er it  is  to  express  the  brightness,  cheerfulness,  and  welcome 
of  a  reception-room,  or  the  tranquillity  of  studious  ease 
which  is  adapted  to  the  library.  If  to  the  walls  be  given 
high  colors,  relief  and  roundness  of  ornamentation,  and 
shade  and  shadow,  instead  of  flat  neutral  tints  of  one  or 
two  colors,  the  walls  are  thereby  apparently  thrust  for- 


24  INDUSTRIAL   DRAWING. 

ward,  the  room  is  made  to  appear  smaller  than  it  is,  and 
the  furniture  is  dwarfed,  and  its  natural  effect  destroyed. 
So,  large  patterns  in  the  carpet  of  a  small  room  produce  a 
like  damaging  effect  upon  both  room  and  furniture,  and 
destroy  that  feeling  of  satisfied  repose  which  is  ever  at- 
tendant upon  true  art.  Vertical  patterns  on  the  walls, 
such  as  columns,  stripes,  etc.,  make  the  walls  appear  higher, 
while  horizontal  lines  and  patterns  lower  the  ceiling. 

PROPOSITION  VIII. — OF  PROPORTION  IN  ORNAMENTATION. 
As  in  every  perfect  tcorJc  of  Architecture  a  true  propor- 
tion will  be  found  to  reign  between  all  the  members  which 
compose  it,  so  throughout  the  decorative  Arts  every  assem- 
blage of  forms  should  be  arranged  on  certain  definite  pro- 
portions /  the  whole  and  each  particular  member  should  be 
a  multiple  of  some  simple  unit. 

Thus  the  height  of  the  Doric  column  was  e'qual  to  six 
times  the  diameter  of  the  lower  end  of  the  shaft ;  the  di- 
ameter of  the  upper  end  of  the  shaft  was  three  fourths  of 
the  diameter  of  the  lower  end ;  and  the  architrave,  frieze, 
and  cornice,  and  all  other  parts,  had  certain  definite  pro- 
portions. In  the  other  orders  the  proportions  were  differ- 
ent;  but  in  each  the  several  parts  were  in  just  proportion 
the  one  to  the  other,  and  to  the  whole.  Nor  were  these 
proportions  arbitrary;  for  they  were  such  as  were  best 
adapted  to  give  expression  to  the  character  of  the  order; 
and  in  no  one  of  these  orders  could  any  important  part  be 
materially  changed  in  its  proportions  without  doing  vio- 
lence to  that  harmony  of  design  which  characterized  the 
entire  structure. 

In  the  infancy  of  Decorative  Art  the  proportions  were 
of  the  most  simple  kind,  in  accordance  with  the  natural  or- 
der of  development.  It  is  the  same  with  the  growth  of 
art  in  individuals.  Thus  as  soon  as  a  child  can  draw  a 
square,  its  first  effort  is  to  divide  it  into  four  squares,  and 
then  into  a  greater  number;  then  to  draw  and  subdivide 
parallelograms ;  then  out  of  the  squares  to  form  lozenge- 
shaped  figures,  etc.,  and  so  on,  as  taste  and  skill  are  devel- 
oped. As  art  advances,  those  proportions  will  be  deemed 


or.:;  A  MENTAL  ACT.  25 

the   most  beautiful  which  the  uneducated  eye   does   net 
readily  detect. 

PROPOSITION  IX. — OF  HARMONY  AND  CONTRAST. 
Where  great  variety  of  form  is  introduced,  harmony  con- 
sists in  the  proper  balancing  and  contrast  of  the  straight, 
the  inclined,  and  the  curved. 

Whether  we  confine  our  attention  to  structural  arrange- 
ment of  edifices,  or  to  decoration  of  surfaces,  there  can  be 
no  perfect  composition  in  which  any  one  of  the  three  pri- 
mary forms  is  wanting.  In  the  Greek  temple  the  straight, 
the  angular  or  inclined,  and  the  curved,  are  in  most  perfect 
relation  to  one  another.  In  the  best  examples  of  Gothic 
architecture  every  tendency  of  lines  to  run  vertically  or 
horizontally  is  immediately  counteracted  by  the  oblique  or 
the  curved.  Thus  the  capping  of  the  buttress  is  exactly 
what  is  required  to  counteract  the  upward  tendency  of  the 


26  INDUSTRIAL   DRAWING. 

straight  lines ;  so  the  gable  contrasts  admirably  with  the 
curved  window-head  and  its  perpendicular  mullions. 

In  surface  decoration  any  arrangement  of  forms,  as  at  A, 
consisting  only  of  straight  lines,  is  monotonous,  and  affords 
but  little  pleasure.  By  introducing  lines  which  tend  to 
carry  the  eye  toward  the  angles,  as  at  B,  the  monotony  is 
broken,  and  the  improvement  is  very  apparent.  Then  add 
lines  giving  a  circular  tendency,  as  at  C,  and  the  eye  reposes 
itself  within  the  outlines  of  the  figure,  and  the  harmony  is 
complete.  In  this  case  the  square  is  the  leading  form  or 
tonic ;  the  oblique  and  curved  forms  are  subordinate. 

An  effect  similar  to  A,  but  an  improvement  upon  it,  is 
produced  by  the  lozenge  composition,  as  at  D.  Add  the 
lines  as  at  E,  and  the  tendency  to  follow  the  oblique  di- 
rection is  corrected ;  but  interpose  the  circles,  as  at  F,  and 
the  eye  at  once  feels  that  repose  which  is  the  result  of  per- 
fect harmony  in  the  combination. 

It  is  owing  to  a  neglect  of  the  principle  here  stated  that 
there  are  so  many  failures  in  wall-paper,  carpets,  oil-cloths, 
and  articles  of  clothing.  The  lines  of  wall-paper  very  gen- 
erally run  through  the  ceiling  most  disagreeably,  because 
the  vertical  is  not  corrected  by  the  inclined,  nor  the  in- 
clined by  the  curved.  So  of  carpets,  the  lines  of  which  fre- 
quently run  in  one  direction  only,  carrying  the  eye  right 
through  the  walls  of  the  apartment.  Many  of  the  checks 
and  plaids  in  common  use  are  objectionable  for  the  same 
reason,  although  a  great  relief  is  sometimes  found  in  their 
coloring. 

PROPOSITION  X. — OF  DISTRIBUTION,  RADIATION,  AND  CON- 
TINUITY. 

In  surface  decoration  by  curvilinear  forms,  all  lines  should 
be  harmoniously  distributed,  and  should  radiate  from  a  par- 
rent  stem  •  and  all  junctions  of  curved  lines  with  curved,  and 
of  curved  lines  with  straight,  should  be  tangential  to  one  an- 
other. 

This  is  a  law  of  the  vegetable  world,  as  seen  in  all  plants 
that  have  curvilinear  forms;  and  Oriental  practice  in  orna- 
mental art  is  in  accordance  with  it. 


ORNAMENTAL  ART.  27 

PROPOSITION  XL  —  OF  CONVENTIONAL  REPRESENTATIONS 
OF  NATURAL  OBJECTS. 

Flowers,  or  other  natural  objects,  should  not  be  used  as  or- 
naments, but,  instead  thereof,  we  should  use  conventional  rep- 
resentations founded  upon  them,  sufficiently  suggestive  to 
convey  the  intended  image  to  the  mind  without  destroying 
the  unity  of  the  object  they  are  employed  to  decorate.  The 
former  is  called  the  NATURALISTIC  style  of  ornamentation, 
the  latter  the  CONVENTIONAL  style. 

Although  this  rule  has  been  universally  obeyed  in  what 
are  deemed  the  classic  periods  of  art,  it  has  been  equally 
violated  when  art  has  been  on  the  decline. 

A  fragile  flower,  or  a  delicate  vine,  carved  in  wood,  stone, 
or  iron,  shocks  our  feeling  of  consistency — an  impropriety 
of  which  the  Egyptians,  the  Greeks,  and  the  early  Romans 
were  never  guilty.  They  made  conventional  representa- 
tions of  natural  objects,  strictly  adhering  to  their  general 
laws  of  form ;  and  hence  their  ornaments,  however  conven- 
tionalized (but  more  especially  those  of  the  ^Egyptians), 
were  always  true  to  nature,  while  they  never,  by  a  too 
servile  imitation  of  the  type,  destroyed  the  consistency  of 
the  representation. 

When  flowers  in  miniature  are  carved  upon  precious 
stones,  or  even  in  iron,  the  delicacy  of  the  workmanship  may 
overcome  our  sense  of  the  unfitness  of  things.  The  flower, 
leaf,  vine,  and  fruit  ornaments  on  vases  and  fruit-dishes  are 
certainly  not  beautiful  except  when  of  diminished  size; 
and  even  then,  if  carved,  they  should  be  executed  in  slight 
relief,  or  merely  etched  in  outline. 

In  contradistinction,  however,  to  the  use  of  color  and 
form  as  mere  accessories  in  industrial  art,  when  we  come  to 
the  fine  art  of  painting,  and  employ  it  to  give  a  representa- 
tion of  real  objects  or  scenes  in  nature,  or  of  those  which 
fancy  creates,  it  is  the  naturalistic  method  which  should 
prevail ;  for  here  the  leading  idea  is  a  faithful  portraiture 
of  what  is  seen  or  imagined ;  and  all  other  ideas  must  be 
subordinate  to  it.  Here  conventionality  of  representation 
would  defeat  the  very  object  in  view. 


III. 

ORNAMENTAL  ART  AMONG  DIFFERENT  NA- 
TIONS, AND  IN  DIFFERENT  PERIODS  OF  CIV- 
ILIZATION. 

I.  ORNAMENT  OF  SAVAGE  TRIBES. 

THE  desire  for  ornament  is  universal,  and  it  increases 
with  all  people  in  the  ratio  of  their  progress  in  civilization. 
Every  where  it  owes  its  origin  to  man's  ambition  to  create 
— to  imitate  the  works  of  the  Creator.  In  the  tattooing  of 
the  human  face  the  savage  strives  to  increase  the  expres- 
sion by  which  he  hopes  to  strike  terror  on  his  enemies  or 
rivals,  or  to  create  what  appears  to  him  a  new  beauty ; 
and  it  is  often  surprising  how  admirably  adapted  are  the 
forms  and  colors  he  uses  to  the  purposes  he  has  in  view. 
After  tattooing  usually  comes  the  formation  of  ornament 
by  painting  or  stamping  patterns  on  the  skins  used  for 
clothing,  or  on  woven  cloths  or  braided  matting.  Then 
follows  the  carving  of  ornaments  on  their  utensils  or  weap- 
ons of  war.  When  the  principal  island  of  the  Friendly 
group  was  first  visited,  one  woman  was  found  to  be  the 
designer  of  all  the  patterns  on  cloths,  matting,  etc.,  in  use 
there ;  and  for  every  new  one,  she  received,  as  a  reward,  a 
certain  number  of  yards  of  cloth. 

What  strikes  us  especially  in  most  ornamental  work  of 
savages,  is  the  adherence  to  that  rule  of  art  which  requires 
a  skillful  balancing  of  the  masses,  whether  of  form  or  color, 
and  a  judicious  correction  of  the  tendency  of  the  eye  to 
run  in  any  one  direction,  by  interposing  lines  that  have  an 
opposite  tendency.  (See  Prop,  ix.) 

Captain  Cook,  noticing  the  extent  to  which  decoration 
was  carried  by  the  Islanders  of  the  Pacific  and  South  Seas, 
speaks  of  their  cloths,  their  basket-work,  their  matting,  etc., 
as  painted  "  in  such  an  endless  variety  of  figures  that  one 


ORNAMENTAL   ABT.  29 

might  suppose  they  borrowed  their  patterns  from  a  mer- 
cer's shop,  in  which  the  most  elegant  productions  of  China 
and  Europe  are  collected,  besides  some  original  patterns 
of  their  own." 

II.  EGYPTIAN  ORNAMENT. 

The  origin  of  art  among  the  Egyptians  is  unknown ;  and 
in  their  architecture,  the  more  ancient  the  monument  the 
more  nearly  perfect  is  the  art.  As  far  back  as  we  can 
trace  their  ornamentation,  a  few  of  the  more  important 
natural  productions  of  the  country  formed  the  basis  of  the 
immense  variety  of  ornament  with  which  the  Egyptians 
decorated  their  temples,  their  palaces,  dress,  utensils,  arti- 
cles of  luxury,  etc. 

All  Egyptian  art  was  symbolic.  Thus  the  lotus  and 
papyrus,  growing  so  luxuriantly  on  the  banks  of  the  Nile, 
the  former  symbolizing  food  for  the  body,  and  the  latter, 
being  used  for  parchment,  and  thus  symbolizing  food  for 
the  mind;  certain  rare  feathers  carried  before  the  king, 
and  thus  symbolizing  royalty ;  the  branches  of  the  palm, 
and  twisted  cord  made  from  its  bark,  etc.  —  such  natural 
products  were  conventionally  used  in  their  decoration ;  and 
hence  they  were  types  of  Egyptian  civilization.  A  lotus 
conventionally  carved  in  stone,  and  forming  a  graceful  ter- 
mination to  a  column,  was  a  fitting  symbol  both  of  plenty  and 
prosperity,  and  of  the  power  of  the  king  over  countries  where 
the  lotus  grew ;  and  thus  the  ornament  added  a  poetic  idea 
to  what  would  otherwise  have  been  but  a  rude  support. 

On  this  basis  of  symbolism,  Egyptian  art  was  of  three 
kinds : 

1st.  Constructive. — Thus,  an  Egyptian  column  represent- 
ed an  enlarged  lotus  or  papyrus  plant,  or  a  bundle  of  such 
plants ;  the  base  representing  the  root,  the  shaft  the  stalk, 
and  the  capital  the  full-blown  flower  surrounded  by  a  bou- 
quet of  smaller  plants  tied  together  by  bands.  And  as 
the  column  represented  one  plant,  three,  or  a  greater  num- 
ber, and  so  of  the  other  parts,  the  variety  of  form  to  which 
the  various  combinations  gave  rise  was  far  beyond  what 
any  other  style  of  architecture  ever  attained  to. 


30  INDUSTRIAL  DRAWING. 

2d.  Representative. — In  their  representation  of  objects,  ev- 
ery thing  —  as  a  flower,  for  example  —  was  portrayed,  not 
as  a  reality,  but  as  an  ideal  representation.  They  did  not 
attempt  to  portray  the  real  flower,  but  something  that 
should  give  the  idea  of  one ;  and  hence  they  adhered  to 
the  principles  of  the  growth  of  plants,  in  the  radiation  of 
leaves,  and  all  veins  on  the  leaves,  in  graceful  curves  from 
the  main  stalk,  or  the  stem.  (See  Prop,  x.)  They  took 
the  general  form  of  the  lotus  plant,  and  changed  it  into 
the  form  of  the  base,  shaft,  and  capital  of  a  column,  while 
it  still  retained  sufficient  resemblance  to  the  lotus  plant 
to  show  whence  the  idea  originated.  And  so  on,  in  all  their 
art  representations.  (See  Prop,  xi.) 

3d.  Decorative. — All  the  paintings  of  the  Egyptians,  pro- 
duced by  few  types,  are  distinguished  by  graceful  sym- 
metry and  perfect  distribution  of  parts.  They  painted  ev- 
ery thing ;  but  using  color  as  they  did  form,  conventionally, 
and  to  distinguish  one  part  from  another,  they  dealt  in  flat 
tints  only,  using  neither  shade  nor  shadow;  and  they  in- 
differently colored  the  leaves  of  the  lotus  green  or  blue. 

III.  ASSYRIAN  AND  PERSIAN  ORNAMENT. 

The  Assyrian  and  Persian  ornaments  which  have  been 
discovered  seem  to  belong  to  a  period  of  decline  in  art,  and 
to  have  been  borrowed,  far  back  in  the  obscurity  of  ages, 
from  an  original  and  more  nearly  perfect  style  —  perhaps 
that  from  which  the  Egyptian  itself  was  derived. 

Assyrian  ornament  is  represented  in  the  same  way  as  the 
Egyptian,  although  it  is  not  based  on  the  same  types ;  and 
indeed  the  natural  types  are  very  few,  Yet  the  natural 
laws  of  radiation  from  the  parent  stem,  and  tangential 
curvature,  are  observed,  although  not  so  strictly  as  in 
Egyptian  art.  In  both  styles,  the  carved  ornaments,  as 
well  as  those  that  were  painted,  are  mostly  in  the  nature 
of  diagrams — geometrial  patterns  being  closely  adhered  to. 
The  little  surface  modeling  that  was  attempted  was  mostly 
in  slight  relief. 


ORNAMENTAL  ART.  31 

IV.  GREEK  ORNAMENT. 

Greek  art,  borrowed  partly  from  the  Egyptian  and  part- 
ly from  the  Assyrian,  gave  to  old  ideas  of  ornament  a  new 
direction.  Rising  rapidly  to  a  high  state  of  perfection,  it 
carried  the  development  of  pure  form  to  a  degree  of  fitness 
and  beauty  which  has  never  since  been  reached ;  and,  from 
the  very  abundant  remains  we  have  of  Greek  ornament, 
we  are  led  to  believe  that  the  presence  of  refined  taste 
among  the  people  was  almost  universal,  and  that  the  land 
was  overflowing  with  artists,  whose  hands  and  minds  were 
so  trained  as  to  enable  them  to  execute  with  unerring 
truth  those  beautiful  ornaments  which  to  this  day  are  the 
great  wonder  of  art.  Greek  ornament,  never  in  profusion 
or  excess,  was  always  strictly  subordinate  to  the  general 
expression  of  the  object  to  which  it  was  affixed. 

But  Greek  art  was  not  symbolical  like  the  Egyptian ;  it 
was  meaningless,  purely  decorative — the  very  embodiment 
of  beauty  for  beauty's  sake — and  hence  wholly  aesthetic ; 
seldom  representative;  and  it  can  hardly  be  said  to  be 
constructive.  The  conventional  rendering  of  natural  ob- 
jects was  so  far  removed  from  the  original  types  as  often 
to  make  it  difficult  to  recognize  any  attempt  at  imitation. 
The  ornament  was  no  part  of  the  construction,  and  was 
thus  unlike  the  Egyptian ;  it  could  be  removed,  and  the 
structure  remain  unchanged.  On  the  Corinthian  capital, 
that  leading  feature  of  Grecian  ornament,  the  acanthus  leaf, 
is  applied,  not  constructed  as  a  part  of  the  edifice.  In  the 
Egyptian,  the  whole  capital,  conventionally  representing 
the  full-blown  flower  of  the  lotus  plant,  is  the  ornament ; 
and  to  remove  any  part  would  destroy  the  whole. 

The  three  great  laws  which  we  find  every  where  in  nat- 
ure— radiation  from  the  parent  stem,  proportionate  distri- 
bution of  the  areas,  and  the  tangential  curvature  of  the 
lines — are  always  obeyed  in  Greek  art,  and  with  so  great  a 
degree  of  perfection  that  the  attempt  to  reproduce  Greek 
ornament  is  rarely  done  with  success. 

There  is  now  little  doubt  that  the  white  marble  temples 
of  the  Greeks  were  entirely  covered  with  painted  ornament. 


32  INDUSTRIAL   DRAWING. 

This  certainly  was  true  as  to  the  ornaments  of  the  mouldings 
on  architrave,  frieze,  and  cornice ;  and  doubtless  the  object 
in  these  cases  was  to  render  the  mouldings  and  carvings 
distinct,  and  make  the  pattern  visible  from  a  distance. 

V.  POMPEIAN  ORNAMENT. 

Pompeii,  a  town  of  Italy,  fifteen  miles  south-east  from 
Naples,  was  destroyed  by  an  eruption  of  Mount  Vesuvius 
in  the  year  A.D.  79.  It  has  since  been  extensively  exca- 
vated, disclosing  the  city  walls,  streets,  temples,  theatres, 
the  forum,  baths,  monuments,  private  dwellings,  domestic 
utensils,  etc.,  the  whole  conveying  the  impression  of  the 
actual  presence  of  a  Roman  town  in  all  the  circumstantial 
reality  of  its  existence  two  thousand  years  ago. 

Pompeian  ornamentation  was  of  two  kinds  —  partly  of 
Grecian  and  partly  of  Roman  origin — but  sufficiently  dis- 
tinct from  either  to  require  a  separate  notice  in  the  history 
of  ornamental  art.  That  derived  from  the  Greek  was  com- 
posed of  conventional  representations  of  objects  in  flat 
tints,  without  shade  or  attempt  at  relief;  the  other,  more 
Roman  in  character,  was  based  mainly  upon  the  acanthus 
scroll,  and  was  interwoven  with  natural  representations  of 
leaves,  flowers,  animals,  etc. — the  germs  of  a  later  Italian 
style  of  ornamentation.  But  the  Pompeian  style  was  ex- 
ceedingly capricious,  beyond  the  range  of  true  art.  The 
Pompeian  pavements  are  the  types  from  which  may  be 
traced  the  immense  variety  of  Byzantine,  Arabian,  and  Mo- 
resque mosaics. 

VI.  ROMAN  ORNAMENT. 

The  temples  of  the  Romans  were  overloaded  with  orna- 
ment ;  and  the  general  proportions  of  Roman  edifices,  and 
the  contours  of  their  moulded  surfaces,  were  entirely  de- 
stroyed by  the  elaborate  surface  modeling  carved  on  them. 
Nor  do  the  Roman  ornaments  grow  naturally  from  the 
surface  like  the  conventional  forms  of  the  Egyptian  capi- 
tal :  they  are  merely  applied  to  it.  The  acanthus  leaves, 
which  by  the  Greeks  were  beautifully  conventionalized, 
were  used  by  the  Romans  with  too  close  an  approxima- 


ORNAMENTAL   ART.  33 

tion  to  nature  :  they  were  also  arranged  iuartistically,  be- 
ing not  even  bound  together  by  the  necking  at  the  top  of 
the  shaft,  but  merely  resting  upon  it.  In  the  Egyptian 
capital,  on  the  contrary,  the  stems  of  the  flowers  round  the 
bell-shaped  capital  being  continued  through  the  necking, 
at  the  same  time  represent  a  beauty  and  express  a  truth. 
The  introduction  of  the  Ionic  volute — a  Grecian  feature — 
into  the  Roman  Composite  order,  fails  to  add  a  beauty,  but 
rather  increases  the  deformity.  The  leaf  ornamentation  of 
the  Romans  adhered  to  the  principle  of  one  leaf  growing 
out  of  another  in  a  continuous  line,  leaf  within  leaf,  and 
leaf  over  leaf— a  principle  very  limited  in  its  application; 
and  it  was  only  in  the  later  Byzantine  period  that  this 
style  began  to  be  abandoned  for  the  true  one  of  a  continu- 
ous stem  throwing  off  ornaments  on  either  side.  Then 
pure  conventional  ornament  began  to  receive  a  new  devel- 
opment. The  true  principle  became  common  in  the  elev- 
enth, twelfth,  and  thirteenth  centuries,  and  is  the  founda- 
tion of  the  Early  English  foliage  style. 

While  Roman  Decorative  Art  abounds  in  the  most  ex- 
quisite specimens  of  drawing  and  modeling,  its  great  de- 
fect consists  in  its  frequent  want  of  adaptation  to  the  pur- 
poses it  was  required  to  fill  as  an  aid  to  the  true  expres- 
sion of  architectural  design.  Roman  decoration,  like  the 
Grecian,  was  strictly  aesthetic — based  on  an  almost  rever. 
ential  regard  for  the  beautiful,  for  beauty's  sake  alone. 

VII.  BYZANTINE  ORNAMENT. 

When  in  the  year  A.D.  328  the  Emperor  Constantine 
transferred  the  seat  of  the  Roman  government  to  Byzan- 
tium (afterward  called  Constantinople,  from  its  founder), 
Roman  art  was  already  in  a  state  of  decline,  or  transforma- 
tion. Constantine  employed  Persian  and  other  Oriental 
artists,  and  artists  from  the  provinces,  in  the  decoration  of 
his  capital ;  and  these  together  soon  began  to  work  a 
change  in  the  traditional  Roman  style,  until  at  length  the 
motley  mass  became  fused  into  one  systematic  whole  dur- 
ing the  long  and  (for  art)  prosperous  reign  of  the  first  Jus- 
tinian. (A.D.  527  to  565.) 

B2 


Si  I3'DUST£IAL   DliAIYI^G. 

Byzantine  art  is  characterized  by  elliptical  curved  out- 
lines, acute-pointed  and  broad-toothed  leaves,  and  thin  con- 
tinuous foliage  springing  from  a  common  stem.  In  sculp- 
ture the  leaves  are  beveled  at  the  edge,  and  deeply  chan- 
neled throughout,  and  drilled,  at  the  several  springings  of 
the  teeth,  with  deep  holes.  Thin  interlaced  patterns  are 
preferred  to  geometrical  designs;  animal  or  other  figures 
are  sparingly  introduced  in  sculpture,  while  in  color  they 
are  principally  confined  to  subjects  of  a  holy  character. 
Rome,  Syria,  Persia,  and  other  countries,  all  took  part  as 
formative  causes  in  the  Byzantine  style  of  art  and  its  ac- 
companying decoration.  The  character  of  the  Byzantine 
school  is  strongly  impressed  on  all  the  earlier  works  of 
Central  and  even  Western  Europe,  which  are  generally 
termed  the  Romanesque  or  Romanized  style,  which  is  con- 
sidered a  fantastic  and  debauched  style  when  applied  to 
architecture.  The  geometrical  mosaic  work  of  Byzantine 
art  belongs  particularly  to  the  Romanesque  period,  espe- 
cially in  Italy.  This  art,  which  flourished  principally  in  the 
twelfth  and  thirteenth  centuries,  consists  in  the  arrange- 
ment of  small  diamond-shaped  pieces  of  glass  into  a  com- 
plicated series  of  diagonal  lines.  Marble  mosaic  work  dif- 
fers from  the  glass  only  in  the  material  used. 

The  influence  of  Byzantine  art  was  all  powerful  in  Eu- 
rope from  the  sixth  to  the  eleventh  century,  and  even  later ; 
and  it  has  served  in  a  great  degree  as  the  basis  of  all  the 
modern  schools  of  decorative  art  in  the  East  and  in  East- 
ern Europe.  , 

VIII.  ARABIAN  ORNAMENT. 

As  every  distinct  form  or  mode  of  civilization  has  been 
characterized  by  its  own  peculiar  style  of  art,  so  when  the 
religion  of  Mohammed  spread  with  astonishing  rapidity 
over  the  East  about  the  middle  of  the  seventh  century,  and 
over  Spain  in  the  early  part  of  the  eighth,  a  new  style  of 
art  arose,  which  gradually  encroached,  in  those  regions,  upon 
the  already  waning  glories  of  the  Byzantine  period. 

Some  of  the  Arabian  mosques  of  Cairo,  erected  in  the 
ninth  century,  remarkable  alike  for  the  grandeur  and  sim- 


OEXAMENTAL   AET.  35 

plicity  of  their  general  forms,  and  the  refinement  and  el- 
egance of  their  decoration,  are  among  the  most  beautiful 
buildings  in  the  world.  Their  elegance  of  ornamentation 
was  probably  derived  primarily  from  the  Persians,  perhaps 
modified  by  Byzantine  influence.  In  their  leafage  orna- 
ments we  observe  traces  of  Greek  origin,  especially  in  the 
modified  form  of  the  acanthus  leaf;  but  they  abandoned 
the  principle  of  leaves  growing  out  one  from  another,  and 
made  the  scroll  continuous  without  break,  while  they  re- 
tained that  universal  principle  of  true  art,  the  radiation  of 
lines  from  a  parent  stem,  and  their  tangential  curvature. 
Like  the  Romans,  they  covered  the  floors  of  their  public 
buildings  with  mosaic  patterns  arranged  on  a  geometrical 
plan ;  but  it  is  surprising  that,  while  the  same  pattern  forms 
of  mosaics  exist  in  Roman,  Byzantine,  Arabian,  and  Moor- 
ish art,  the  general  style  of  each  differs  widely  from  all  the 
others.  It  is  like  the  same  idea  expressed  in  four  differ- 
ent languages.  The  twisted  cord,  the  interlacing  of  lines 
straight  or  curved,  the  crossing  and  interlacing  of  two 
squares,  and  the  equilateral  triangle  within  a  hexagon,  are 
the  starting-points  in  each. 

What  is  called  Arabesque  ornament  consists  of  a  fanciful, 
capricious,  and  ideal  mixture  of  all  sorts  of  figures  of  men 
and  animals,  both  real  and  imaginary;  also  all  sorts  of 
plants,  fruit,  and  foliage,  involved  and  twisted,  and  upon 
which  the  animals  and  other  objects  rest.  The  Arabians 
did  not  originate  this  style,  although  it  is  named  from 
them ;  and  in  pure  Arabesque,  figures  of  animals  are  ex- 
cluded, as  they  were  forbidden  by  the  Koran. 

It  is  strange  that  while  the  Arabians  have  left  traces  of 
fine  Saracenic  art  all  through  Northern  Africa,  and  in  Spain, 
scarcely  a  vestige  of  it  can  now  be  found  in  their  native 
country,  Arabia. 

IX.  TURKISH  ORXAMEXT. 

Although  the  Turks  and  the  Arabians  have  the  same  re- 
ligion, yet,  being  of  different  national  origin,  their  art  rep- 
resentations are,  as  might  be  expected,  somewhat  different. 
The  architecture  of  the  Turks,  as  seen  at  Constantinople, 


36  INDUSTRIAL   DBA  WING. 

is  mainly  based  upon  the  early  Byzantine  monuments,  ex- 
cept their  modern  edifices,  which  are  designed  in  the  most 
European  style.  Their  system  of  ornamentation  is  of  a 
mixed  character  —  Arabian  and  Persian  floral  ornaments 
being  found  side  by  side  with  debased  Roman  and  Renais- 
sance details.  The  art  instinct  of  the  Turks  is  quite  in- 
ferior to  that  of  the  East  Indians.  The  only  good  exam- 
ples we  have  of  Turkish  ornamentation  is  in  Turkey  car- 
pets ;  and  these  are  chiefly  executed  in  Asia  Minor,  and 
most  probably  not  by  Turks.  The  designs  are  thoroughly 
Arabian.  The  Turk  is  unimaginative. 

X.  MORESQUE  OR  MOORISH  ORNAMENT. 

In  the  ornamental  art  of  the  Moors,  who  established  the 
seat  of  their  power  in  Spain  during  the  eighth  century,  we 
have  another  illustration  of  the  results  produced  by  corre- 
sponding influences  of  religious  faith  and  diversities  of  na- 
tional character.  The  main  differences  between  the  Ara- 
bian and  Moorish  edifices  consist  in  this :  that  the  former 
are  distinguished  most  for  their  grandeur,  the  latter  for 
their  refinement  and  elegance.  In  ornamentation  the  Moors 
were  unsurpassed ;  and  in  it  they  carried  out  the  princi- 
ples of  true  art,  even  beyond  the  attainments  of  the  Greeks 
themselves. 

Arabian  and  Moorish  art  were  alike  wanting  in  symbol- 
ism; but  the  Moors  compensated  for  this  want  by  the 
beauty  of  their  ornamental  written  inscriptions,  and  the 
nobleness  of  the  sentiments  they  expressed.  To  the  artist 
these  inscriptions  furnished  the  most  exquisite  lessons  in 
art;  to  the  people  they  proclaimed  the  might,  majesty, and 
good  deeds  of  the  king ;  and  to  the  king  they  never  ceased 
to  declare  that  there  was  none  powerful  but  God ;  that  He 
alone  was  conqueror,  and  that  to  Him  alone  was  ever  due 
praise  and  glory.  A  law  of  the  Mohammedan  religion  for- 
bade the  representation  of  animals,  or  of  the  human  figure. 

In  the  best  specimens  of  Moorish  architecture  the  deco- 
ration always  arises  naturally  from  the  construction ;  and, 
although  every  part  of  the  surface  may  be  decorated,  there 
is  never  a  useless  or  a  superfluous  ornament.  All  lines 


OEXAMEXTAL   AET.  37 

o-row  out  of  one  another  in  natural  undulations,  and  every 
ornament  can  be  traced  to  its  branch  or  root ;  and  there  is 
no  such  thing  as  an  ornament  just  jotted  down  to  fill  a 
space,  without  any  other  reason  for  its  existence. 

The  best  Moorish  ornamentation  is  found  in  the  Alham- 
bra,  a  celebrated  palace  of  the  Moorish  kings,  at  Granada, 
in  Spain.  This  immense  and  justly  famous  structure,  of 
rather  forbidding  exterior,  but  gorgeous  within  almost  be- 
yond description,  was  erected  in  the  thirteenth  century; 
and  much  of  it  remains  perfect  at  the  present  day.  It  has 
been  said  by  a  competent  judge  that  "Every  principle 
which  we  can  derive  from  the  study  of  the  ornamental  art 
of  any  other  people  is  not  only  ever  present  here,  but  was 
by  the  Moors  more  universally  and  truly  obeyed."  And 
further,  that  "  We  find  in  the  Alhambra  the  speaking  art 
of  the  Egyptians,  the  natural  grace  and  refinement  of  the 
Greeks,  and  the  geometrical  combinations  of  the  Romans, 
the  Byzantines,  and  the  Arabs."  The  walls  of  the  Alham- 
bra were  covered  with  a  profusion  of  ornamentation,  which 
had  the  appearance  of  a  congeries  of  paintings,  incrusta- 
tions, mosaics,  gilding,  and  foliage ;  and  nothing  could  be 
more  splendid  and  brilliant  than  the  effects  that  resulted 
from  their  combinations.  The  mode  of  piercing  the  domes 
for  light,  by  means  of  star-like  openings,  produced  an  al- 
most magical  effect. 

XI.  PERSIAN  ORNAMENT. 

The  Mohammedan  architecture  of  Persia,  and  Persian  or- 
namentation, are  alike  a  mixed  style,  and  are  far  inferior  to 
the  Arabian,  as  exhibited  in  the  buildings  at  Cairo.  The 
Persians,  unlike  the  Arabs  and  the  Moors,  mixed  up  the 
forms  of  natural  flowers  and  animal  life  with  conventional 
ornament. 

XII.  EAST -INDIAN  ORNAMENT. 

Numerous  manufactures  calculated  to  give  a  high  idea 
of  the  ingenuity  and  taste  of  the  people  of  British  India 
appeared  in  the  Great  Exhibition  of  the  Industry  of  all  Na- 
tions, in  London,  in  1851.  Among  these  were  various  ar- 


38  INDUSTRIAL   DRAWING. 

tides  in  agate  from  Bombay,  mirrors  from  Lahore,  marble 
chairs  from  Ajmeer,  embroidered  shawls,  scarfs,  etc.,  from 
Cashmere,  carpets  from  Bangalore,  and  a  variety  of  articles 
in  iron  inlaid  with  silver.  In  the  application  of  art  to  man- 
ufactures the  East  Indians  exhibit  great  unity  of  design, 
and  skill  and  judgment  in  the  application,  with  great  ele- 
gance and  refinement  in  the  execution.  In  these  respects 
they  seem  far  to  surpass  the  Europeans,  who,  says  Mr. 
Owen  Jones,  "in  a  fruitless  struggle  after  novelty,  irrespec- 
tive of  fitness,  base  their  designs  upon  a  system  of  copying 
and  misapplying  the  received  forms  of  beauty  of  every  by- 
gone style  of  art."  All  the  laws  of  the  distribution  of  form 
which  are  observed  in  the  Arabian  and  Moresque  orna- 
'ments  are  equally  to  be  found  in  the  productions  of  India, 
while  the  coloring  of  the  latter  is  said  to  be  so  perfectly 
harmonized  that  it  is  impossible  to  find  a  discord.  This,  of 
course,  refers  to  the  selected  articles  placed  on  exhibition 
in  1851. 

XIII.  HINDOO  ORNAMENT. 

We  have  but  little  reliable  information  about  the  an- 
cient, or  Hindoo,  architecture  of  India ;  yet  we  know  this 
much,  that  the  Hindoos  had  definite  rules  of  architectural 
proportion  and  symmetry.  One  of  their  ancient  precepts, 
quoted  by  a  modern  writer,  says,  u  Woe  to  them  who  dwell 
in  a  house  not  built  according  to  the  proportions  of  sym- 
metry. In  building  an  edifice,  therefore,  let  all  its  parts, 
from  the  basement  to  the  roof,  be  duly  considered." 

The  architectural  features  of  Hindoo  buildings  consist 
chiefly  of  mouldings  heaped  up  one  over  the  other.  There 
is  very  little  marked  character  in  their  ornaments,  which 
are  never  elaborately  profuse,  and  which  show  both  an 
Egyptian  and  a  Grecian  influence. 

XIV.  CHINESE  ORNAMENT. 

Notwithstanding  the  great  antiquity  of  Chinese  civiliza- 
tion, and  the  perfection  reached  in  their  manufacturing  pro- 
cesses ages  before  our  time,  the  Chinese  do  not  appear  to 
have  made  much  advance  in  the  fine  arts.    They  show  very 


OEXAMENTAL   AET.  39 

little  appreciation  of  pure  form,  beyond  geometrical  pat- 
terns ;  but  they  possess  the  happy  instinct  of  harmonizing 
colors.  Their  decoration  is  of  a  very  primitive  kind.  The 
Chinese  are  totally  unimaginative ;  and  their  ornamentation 
is  a  very  faithful  expression  of  the  nature  of  this  peculiar 
people — oddness. 

XV.  CELTIC  ORNAMENT. 

The  Celts — the  early  inhabitants  of  the  British  Isles — had 
a  style  of  ornamentation  peculiarly  their  own,  and  singu- 
larly at  variance  with  any  thing  that  can  be  found  in  any 
other  part  of  the  world.  Celtic  ornament  was  doubtless  of 
independent  origin,  but  it  every  where  bears  the  impress 
received  by  the  early  introduction  of  Christianity  into  the 
islands. 

The  chief  peculiarities  of  Celtic  ornament  consist,  first,  in 
the  entire  absence  of  foliage  or  other  vegetable  ornament ; 
and,  secondly,  in  the  extreme  intricacy  and  excessive  mi- 
nuteness and  elaboration  of  the  various  patterns,  mostly 
geometrical,  consisting  of  interlaced  ribbon-work ;  diago- 
nal, straight,  or  spiral  lines ;  and  strange,  monstrous  ani- 
mals or  birds,  with  their  tail-feathers,  top-knots,  and  tongues 
extended  into  long  interlacing  ribbons,  which  were  inter- 
twined in  almost  endless  forms,  and  in  the  most  fantastic 
manner.  Celtic  manuscripts  of  the  Gospels  were  often  orna- 
mented with  a  great  profusion  of  these  intricate  designs. 

What  is  called  the  Celtic  ornamentation  was  practiced 
throughout  Great  Britain  and  Ireland  from  the  fourth  or 
fifth  to  the  tenth  or  eleventh  centuries.  There  was  a  later 
Anglo-Saxon  ornamentation,  equally  elaborate,  employed  in 
the  decoration  of  manuscripts  of  the  Gospels  and  other  holy 
writings ;  but  here  leaves,  stems,  birds,  etc.,  were  intro- 
duced, and  interwoven  with  gold  bars,  circles,  squares,  loz- 
enges, quarterfoils,  etc. 

XVI.  MEDIAEVAL  OK  GOTHIC  ORNAMENT. 
The  high-pitched  gable  and  the  pointed  arch,  with  a  con- 
sequent slender  proportion  of  towers,  columns,  and  capitals, 
are  the  leading  characteristics  of  medieval  or  Gothic  archi- 


40  INDUSTRIAL   DRAWING. 

lecture,  which  came  into  general  use  in  Europe  in  the  thir- 
teenth century.  Mediaeval  Gothic  art,  like  the  Egyptian, 
was  symbolic,  deriving  its  types  from  the  prevailing  religious 
ideas  of  the  period.  Thus  the  churches  and  the  cathedrals 
of  the  Middle  Ages  were  built  in  the  form  of  a  cross — the  sign 
and  symbol  of  the  Christian  faith.  The  numbers  three,  five, 
and  seven,  denoting  the  Trinity,  the  five  traditional  wounds 
of  the  Saviour,  and  the  seven  Sacraments,  were  preserved 
as  emblematical  in  the  nave  and  two  aisles,  in  the  trefoiled 
arches  and  windows,  in  the  foils  of  the  tracery,  and  in  the 
seven  leaflets  of  the  sculptured  foliage ;  while  the  narrow- 
pointed  arches,  and  the  numerous  finger-like  pinnacles,  ris- 
ing above  the  gloom  of  the  dimly  lighted  place  of  worship, 
symbolized  the  faith  which  pointed  the  soul  upward  from 
the  trials  of  earth  to  the  happy  homes  of  the  redeemed. 
The  transition  from  the  Romanesque  (later  Roman)  or 
rounded  style  to  the  pointed  is  easily  traced  in  the  numer- 
ous buildings  in  which  the  two  styles  are  intermingled  ;  but 
the  passage  from  Romanesque  ornament  to  Gothic  is  not  so 
clear.  In  the  latter,  new  combinations  of  ornaments  and 
tracery  suddenly  arise.  The  piercings  for  windows  be- 
come clustered  in  groups,  soon  to  be  moulded  into  a  net- 
work of  enveloping  tracery;  the  acanthus  leaf  disappears; 
in  the  capitals  of  columns  of  pure  Gothic  style,  the  orna- 
ment arises  directly  from  the  shaft,  which,  above  the  neck- 
ing, is  split  into  a  series  of  stems,  each  terminating  in  a 
conventional  flower — the  whole  being  quite  analogous  to 
the  Egyptian  mode  of  decorating  the  capital. 

In  the  interior  of  the  early  Gothic  buildings  every  mould- 
ing had  its  color  best  adapted  to  develop  its  form;  and 
from  the  floor  to  the  roof  not  an  inch  of  space  but  had  its 
appropriate  ornament,  the  whole  producing  an  effect  grand 
almost  beyond  description.  But  so  suddenly  did  this  pro- 
fuse style  of  ornament  attain  its  perfection,  that  it  almost 
immediately  began  to  decline.  What  is  called  ornamental 
illumination,  that  is,  the  decoration  of  writing  by  means  of 
colors,  and,  especially,  the  decoration  of  the  initial  letters  to 
pages  of  manuscript,  attained  a  high  degree  of  perfection 
under  the  influence  of  the  Gothic  style. 


OEXAMENTAL  ART.  41 

While  Gothic  ornamentation  retained  its  conventional 
character,  there  was  a  boundless  field  for  development : 
when  it  became  a  mere  imitation  of  natural  objects,  and  rep- 
resented stems,  flowers,  insects,  etc.,  true  to  life,  all  ideality 
fled,  and  there  could  be  no  further  progress  in  the  art. 

XVII.  RENAISSANCE  ORNAMENT. 

The  fact  that  the  soil  of  Italy  was  so  covered  with  the  re- 
mains of  Roman  greatness  that  it  was  impossible  for  the 
Italians  to  forget  them,  however  they  might  neglect  the  les- 
sons they  were  calculated  to  teach,  was  probably  the  rea- 
son why  Gothic  art  took  but  little  root  in  Italy,  where  it 
was  ever  regarded  as  of  barbarian  origin.  When,  in  the  fif- 
teenth century,  classical  learning  revived  in  Italy,  and  the 
art  of  printing  disseminated  its  treasures,  a  taste  for  classic 
art  revived  also;  and  the  style  of  ornamentation  to  which 
it  gave  rise,  formed  upon  classic  models,  is  called  Renais- 
sance ornament ;  and  the  period  of  its  glory  the  Restora- 
tion, or  Renaissance  period. 

A  combination  of  architecture  and  decorative  sculpture 
was  a  distinguishing  feature  of  the  Renaissance  style.  Fig- 
ures, foliage,  and  conventional  ornaments  were  so  happily 
blended  with  mouldings,  and  other  structural  forms,  as  to 
convey  the  idea  that  the  whole  sprung  to  life  in  one  perfect 
form  in  the  mind  of  the  artist  by  whom  the  work  was  ex- 
ecuted. To  Raphael  (early  in  the  sixteenth  century),  both 
sculptor  and  painter,  we  owe  the  most  splendid  specimens 
of  the  Arabesque  style,  which  he  dignified,  and  left  with 
nothing  more  to  be  desired.  (See  Arabian  Ornament.)  Ara- 
besques lose  their  character  when  applied  to  large  objects; 
neither  are  they  appropriate  where  gravity  of  style  is  re- 
quired. 

All  the  great  painters  of  Italy  were  ornamental  sculptors 
also.  Their  sculptured  ornaments  were  ingeniously  arranged 
on  different  planes,  instead  of  on  one  uniform  flat  surface, 
so  as  best  to  show  the  diversities  of  light  and  shade.  Much 
of  the  splendid  painting  done  by  the  Italian  masters,  from 
Giotto  to  Raphael — from  the  year  1290  to  1520 — was  mu- 
ral decoration,  now  generally  called  fresco.  In  true  fresco. 


42  INDUSTRIAL   DRAWING. 

the  artist  incorporated  his  colors  with  the  plaster  before  it 
was  dry,  by  which  the  colors  became  as  permanent  as  the 
wall  itself.  This  kind  of  painting  was  so  clear  and  trans- 
parent, and  reflected  the  light  so  well,  as  to  be  peculiarly 
suited  to  the  interior  of  dimly  lighted  buildings ;  and  it  is 
said  that  the  eye  which  has  been  accustomed  to  look  upon 
it  can  scarcely  be  reconciled  to  oil  pictures.  It  is  a  well- 
known  saying  of  Michael  Angelo,  that  fresco  is  fit  for  men, 
oil  painting  for  women,  and  the  luxurious  and  idle. 

XVIII.  ELIZABETHAN  ORNAMENT. 

The  revival  of  art  in  Italy  soon  spread  over  France  and 
Germany,  and  about  the  year  1520  extended  into  England, 
where  it  soon  triumphed  over  the  late  Gothic  style.  The 
true  Elizabethan  period  of  art  embraced  only  about  a  cent- 
ury. It  is  simply  a  modification  of  foreign  models,  and  has 
little  claim  to  originality. 

The  characteristics  of  Elizabethan  ornament  may  be  de- 
scribed as  consisting  chiefly  of  a  grotesque  and  complicated 
variety  of  pierced  scroll-work,  with  curled  edges;  interlaced 
bands,  sometimes  on  a  geometrical  pattern,  but  generally 
flowing  and  capricious;  curved  and  broken  outlines;  fes- 
toons, fruit,  and  drapery,  interspersed  with  roughly  exe- 
cuted figures  of  human  beings ;  grotesque  monsters  and  ani- 
mals, with  here  and  there  large  and  flowing  designs  of  nat- 
ural branch  and  leaf  ornament ;  rustic  ball  and  diamond 
work ;  paneled  compartments,  often  filled  with  foliage,  or 
coats  of  arms,  etc.,  etc. :  the  whole  founded  on  exaggerated 
models  of  the  early  Renaissance  school.  By  the  middle  of 
the  seventeenth  century  the  more  marked  characteristics 
of  the  Elizabethan  style  had  completely  died  out. 


MODERN  ORNAMENTAL  ART. 

There  is,  no  doubt,  a  very  decided  tendency  in  modern 
ornamental  art  to  copy  natural  forms  as  faithfully  as  pos- 
sible for  all  decorative  purposes.  We  see  this,  alike,  in  our 
floral  carpets,  floral  wall-papers,  floral  curtains,  and  in  the 


ORNAMENTAL   AET.  43 

floral  carvings  of  our  structures  of  wood,  stone,  and  iron. 
Yet  when  perfection  shall  have  been  attained  in  this  mode 
of  ornamentation — if  it  has  not  been  already — and  which  is 
but  the  mere  copying  of  nature,  and  devoid  of  all  original- 
ity of  design,  how  little  has  the  artist  accomplished  in  the 
development  and  application  of  art  principles,  and  what  fur- 
ther can  he  attain  to  ? 

But  when,  on  the  contrary,  the  progress  of  true  art  shall 
be  acknowledged  to  lie  in  the  direction  of  idealizing  the 
forms  of  nature — giving  to  them  a  conventional  represen- 
tation while  adhering  to  the  principles  of  natural  growth,  in 
the  manner  in  which  art  grew  up  among  the  Egyptians  and 
the  Greeks — the  artist  will  be  left  free  to  follow  the  bent  of 
his  genius,  and  to  select  from,  and  conventionally  adopt, 
whatever  natural  forms  he  may  find  best  suited  to  his  pur- 
poses. Then  there  may  be  advance  in  art  beyond  the  copy- 
ing and  intermingling  of  those  olden  styles,  which  now  ex- 
cite in  us  but  little  sympathy ;  but  until  then  we  shall  prob- 
ably rest  content  in  the  idea  that  all  available  modes  and 
forms  have  been  used  by  those  who  preceded  us,  and  that 
there  are  no  untrodden  domains  of  art  left  for  us  to  explore. 


PART  II. 
PRINCIPLES  AND  PRACTICE 


OP 


INDUSTRIAL  DRAWING. 


DRAWING-BOOK  No.  I. 


I.  MATERIALS  AND  DIRECTIONS. 

1.  FOB  PAPER  to  draw  on,  use  "  Wittson's  Cabinet  Draio- 
ing -Paper"  for  Drawing-Books  Nos.  I.,  IL,  III.,  and  IV. 
This  paper  is  printed  in  fine  red  or  pink  lines,  to  correspond 
to  the  ruling  in  the  Drawing-Books ;  and  it  has  a  border  so 
ruled  and  lettered  as  to  furnish  convenient  guides  for  the 
accurate  drawing  of  the  diagonal  and  serai-diagonal  lines 
of  Cabinet  Perspective,  as  illustrated  in  the  Second,  Third, 
and  Fourth  Drawing-Books.     Of  this  drawing-paper,  No.  1 
is  the  same  in  size  as  the  pages  in  the  Drawing-Books,  and 
No.  2  is  four  times  the  size.    There  is  also  "  Isometrical 
Drawing- Paper  No.  1,"  of  the  same  size  as  the  No.  1  Cab- 
inet Drawing-Paper,  for  use  in  isometrical  drawings,  as  il- 
lustrated in  the  Appendix  to  this  volume.     The  fine  pink 
lines  of  the  drawing -paper  do  not  in  the  least  interfere 
with  the  pencil  drawings. 

2.  For  PENCILS,  use  Faber's  Nos.  1,  2,  3,  and  4,  which  are 
round  black  pencils.     No.  4  being  the  hardest  of  these,  is 
used  for  fine,  hard  lines  only,  or  very  light  shading ;  No.  3 
for  common  outline  drawing  and  shading ;  and  No.  2  for 
heavy  and  distinct  dark  lines  and  edges.     No.  1,  very  dark 
and  soft,  is  little  used.     There  are  also  very  superior  Fa- 
ber  pencils,  of  light  wood,  hexagonal  in  form,  and  numbered 
by  letters  H,  HH,  HHH,  and  HHHH :  H  being  soft  pencils, 
and  HHHH  very  hard  and  fine. 

There  are  also  what  are  called  the  Eagle  pencils— the  H 
pencil  for  light  shading  and  lines,  and  the  F  for  common 
shading.  The  common  Eagle  pencils  marked  1,  2,  and  3, 
are  of  inferior  grade. 


48  INDUSTRIAL   DRAWING.  [BOOK   NO.  I. 

3.  For  most  industrial  drawings,  however,  India  ink  is 
more  convenient,  and  better,  for  shading,  than  the  pencil. 
A  cake  of  good  India  ink,  about  two  inches  long,  that  will 
go  further  in  shading  than  a  hundred  pencils,  may  be  bought 
of  almost  any  bookseller  or  stationer  for  some  twenty  or 
thirty  cents.     Two  or  three  camel's-hair  pencils  (or  brush- 
es) will  also  be  needed.     Price,  three  or  four  cents  each. 
To  use  the  ink,  put  half  a  teaspoonful,  or  less,  of  water  in  a 
small  saucer  (or  the  smallest  china  plate,  about  two  inches 
in  diameter;  or  a  small  glass  salt-cellar  is  better),  and  rub 
one  end  of  the  India-ink  cake  in  it,  giving  the  water  the 
depth  of  tint  that  is  required.     With  one  of  the  brushes 
flow  the  ink  over  those  portions  of  the  drawing  that  are  to 
be  shaded.     When  the  ink  is  dry,  apply  the  wash  a  second 
time  to  those  portions  that  require  a  darker  shade  than  the 
lighter  portions,  and  apply  it  a  third  time  to  those  portions, 
if  any,  that  require  a  still  darker  shade.     In  this  manner 
any  required  depth  of  even  shade  may  be  given.     Be  care- 
ful and  not  make  the  ink  too  dark  at  first ;  and,  as  it  dries 
up  quite  rapidly  in  the  saucer,  water  must  be  supplied  from 
time  to  time  to  keep  it  of  a  uniform  tint.     It  produces  a 
good  eifect  to  first  wash  lightly,  with  India  ink,  those  por- 
tions of  a  drawing  that  require  shading ;  and  then,  when 
the  ink  is  dry,  to  put  on  the  line  shading  with  the  pencil. 

4.  For  many  of  the  curvilinear  drawings,  in  which  parts 
or  wholes  of  perfect  circbs  are  used,  a  pair  of  compasses 
adapted  to  receive  a  pencil  will  be  needed ;  or,  what  will 
answer  the  purpose  very  well,  a  pencil  may  be  split  and 
tied  firmly  to  one  of  the  legs  of  a  pair  of  ordinary  brass 
compasses  or  dividers. 

5.  A  ruler  will  also  be  needed  for  drawing  long  straight 
lines.     It  should  be  beveled  oif  on  one  side  to  a  very  thin 
edge.     A  ruler  with  one  thin  metallic  edge  is  the  most  con- 
venient. 

6.  For  the  purpose  of  Blackboard  Exercises  in  connection 
with  drawing  on  paper,  every  school  in  which  industrial 
drawing  is  taught  should  be  provided  with  a  blackboard 
of  convenient  size,  having  fine  red  lines  painted  on  it,  both 
vertically  and  horizontally,  at  right  angles  to  one  another, 


MATERIALS    AND   i>I2ECTIOXS.  49 

and  two  inches  apart.  Any  careful  painter  «an  prepare  a 
board  in  this  manner.  The  board  should  not  be  varnished. 
The  red  lines  drawn  on  the  board  will  interfere  very  little 
with  the  use  of  the  board  for  ordinary  purposes.  Tho 
school  should  also  be  provided  with  one  pair  of  chalk-cray- 
on compasses,  for  the  drawing  of  regular  curves  on  the 
blackboard.  Any  ingenious  carpenter  can  make  a  pair  that 
will  answer  very  well.  One  of  the  points  may  be  hollowed 
out  to  receive  the  crayon,  which  may  be  tied  in. 

7.  All  the  figures  in  a  lesson,  or  on  a  page  of  the  Drawing- 
Books,  should  be  first  copied  by  the  pupils  on  the  lined 
drawing-paper,  and  then  the  accompanying  Problems  should 
be  drawn,  and  then  the  free-hand  blackboard  exercises, 
when  such  are  suggested.     The  pupils  should  also  explain 
the  drawings  fully — their  measurements  according  to  the 
scale  given  on  the  paper,  and  their  real  measures  when 
drawn  on  the  blackboard.     But  if  any  of  the  pupils  are 
too  young  to  understand  the  few  elementary  principles  of 
surface  measurement  that  are  given  in  Drawing-Book  No.  I., 
these  principles  may  be  passed  over  for  the  present,  as 
they  will  come  up  again  in  a  more  extended  exposition  of 
the  Drawing,  Measurement,  and  Relations  of  Surfaces  and 
Solids. 

8.  Although  free-hand  drawing  can  be  carried  out  in  the 
present  series  quite  as  extensively  as  in  any  other  series, 
and  perhaps  with  more  effect  than  in   any  other,  as  the 
guide-lines  at  once  detect  all  inaccuracies;  and  although 
this  kind  of  preliminary  practice  is  important  for  all  de- 
signers in  art,  and  especially  for  artists  by  profession,  yet 
we  would  remind  teachers  and  pupils  that  it  is  never  re- 
lied on  by  architects,  draughtsmen,  and  artisans  for  the 
drawing  of  working-patterns  or  designs  for  industrial  pur- 
poses, and  that  most  of  the  copies  which  are  given  in  the 
drawing-books  for  practice  in  free-hand  drawing  are  there 
executed,  with  elaborate  care,  by  the  aid  of  ruler  and  com- 
pass.    Even  the  best  of  artists  do  not  hesitate  to  resort  to 
all  possible  mechanical  appliances  by  which  their  work  can 
be  improved  ;  and  it  would  be  strange,  indeed,  if  we  should 
deny  to  children  those  aids  which  we  allow  to  age  and  ex- 

C 


50  INDUSTRIAL   DBA  WING.  [BOOK   NO.  I. 

perience.  While,  therefore,  we  recommend  free-hand  draw- 
ing in  elementary  exercises,  and  also  in  all  portions  of 
copies  or  original  designs  which  can  be  well  executed  there- 
by, we  would  advise  advanced  pupils  to  make  use  of  all 
other  aids  that  are  essential  to  accuracy  of  result.  Fre- 
quent directions  are  given  throughout  the  work  for  free- 
hand exercises  in  drawing  on  the  blackboard. 

9.  For  the  purpose  of  getting  the  full  cfiect  of  a  drawing 
in  diagonal  Cabinet  Perspective  (Books  II.,  III.,  and  IV.), 
partially  close  the  hand,  and  through  the  tubular  opening 
thus  formed  look  at  the  drawing  from  a  position  a  little 
above  and  at  the  right  of  it.     On  thus  viewing  it  intently 
for  half  a  minute,  the  drawing  will  seem  to  stand  out  in 
bold  relief  from  the  paper;  and  if  there  are  any  inaccura- 
cies in  the  perspective,  they  will  be  readily  detected  by 
the  unnatural  apJ3earances  which  they  will  thus  be  made 
to  present. 

10.  If  the  teacher  should  find  some  few  slight  inaccura- 
cies in  which  the  diagrams  in  the  Drawing-Books  do  not 
fully  come  up  to  the  descriptions  of  them,  they  must  at- 
tribute it  to  the  occasional  want  of  care  in  the  artists  who 
copied  them  from  the  original  drawings.     The  errors,  how- 
ever, are  believed  to  be  few,  and  of  little  importance ;  and 
the  teacher  who  gets  hold  of  the  principles  will  easily  cor- 
rect them. 

11.  It  should  be  remarked,  also,  that  drawings  in  pencil 
and  India  ink,  if  well  executed,  and  especially  if  made  on 
the  pink-ruled  drawing-paper,  will  be  clearer  in  shading, 
more  distinct  in  outline,  and  will  show  to  better  advantage 
generally,  than  those  in  the  Drawing-Books. 

ElSir"  12.  For  convenience  of  adapting  the  explanations  of 
drawings  given  in  the  Drawing-Books  to  those  made  on 
the  blackboard,  let  it  be  understood  that  the  lines  on  the 
blackboard  are  in  all  cases  (unless  otherwise  directed)  sup- 
posed to  be  drawn  to  the  same  scale  as  those  assigned  for 
the  lines  of  the  printed  drawings. 


STRAIGHT   LIXES    AND   PLANE    SURFACES.  51 


H.  STRAIGHT  LINES  AND  PLANE    SURFACES. 
PAGE  ONE. 

LESSON  I.  Horizontal  Parallel  Lines. — A  horizontal  line 
is  a  line  that  has  all  its  points  equally  high,  or  on  a  level 
with  the  horizon.  Parallel  lines  are  lines  that  extend  in  the 
same  direction,  and  that  are  equally  distant  from  one  anoth- 
er, however  far  they  may  be  extended.  Thus,  the  lines  that 
cross  the  paper  from  left  to  right  are  parallel  lines,  one 
eighth  of  an  inch  apart ;  and  they  are  also  horizontal  lines 
when  the  paper  lies  flat  upon  the  table,  and  also  when  it  is 
raised  to  an  upright  position.  All  the  lines  in  Lesson  I. 
may  be  considered  horizontal  and  parallel. 

In  drawing  the  copies  on  this  page,  use  a  No.  3  or  No.  2 
pencil,  rounded  at  the  point,  and  not  sharp.  Use  no  ruler. 
In  figure  No.  1,  draw  all  the  lines  on  the  fine-ruled  horizontal 
red  lines  seen  on  the  drawing-paper — first  tracing  each  line 
very  lightly,  carrying  the  pencil  a  part  of  the  time  from  left 
to  right,  and  a  part  of  the  time  from  right  to  left,  so  as  to 
acquire  a  free  command  of  the  hand.  Finish  by  drawing 
each  line  firm  and  distinct,  and  as  true  and  even  as  possible. 
In  the  first  column  the  lines  are  one  eighth  of  an  inch  long; 
in  the  second  column  two  eighths,  or  one  quarter  of  an 
inch ;  and  in  the  third  column  three  eighths  of  an  inch  long. 
The  printed  vertical  and  horizontal  lines  in  the  Drawing- 
Book,  and  also  on  the  drawing-paper,  are  one  eighth  of  an 
inch  apart. 

In  No.  I.,  the  pencil  lines  are  drawn  on  the  ruled  lines,  one 
eighth  of  an  inch  apart ;  in  No.  II.,  they  are  first  drawn  the 
same  as  in  No.  I.,  and  then  a  line  is  drawn  between  every 
two;  in  No.  III.,  two  lines  are  drawn  equally  distant  between 
every  two  lines  first  drawn  as  in  No.  I.  No.  III.  represents 
coarse  shading.  Let  the  pupil  imitate  the  foregoing  with 
free-hand  drawing  on  the  red-lined  blackboard,  and  tell  the 
lengths  of  the  lines  thus  drawn — as  two  inches  four  inches, 
six  inches,  etc. ;  and  their  distances  apart. 


52  INDUSTRIAL   DRAWING.  [BOOK   NO.  I. 

LESSON  II.  Vertical  Parallel  Lines.  —  A  vertical  line  is 
one  that  is  exactly  upright  in  position — such  a  line  as  that 
which  is  formed  by  suspending  a  weight  by  a  string.  The 
lines  in  Lesson  II.  represent  vertical  lines ;  but  they  are 
really  vertical  only  when  the  paper  is  placed  in  an  upright 
position,  and  with  the  heading  of  the  page  upward.  These 
vertical  lines  are  parallel,  for  the  same  reason  that  those  in 
Lesson  I.  are  parallel. 

Draw  the  lines  in  Lesson  II.  from  the  top  downward, 
first  going  over  each  line  lightly,  once  or  twice ;  and,  when 
the  line  is  accurately  traced  from  point  to  point,  finish  by 
marking  it  firmly. 

What  are  the  respective  lengths  of  the  lines  in  No.  1  ?  In 
No.  3?  In  No.  4? 

In  Nos.  2  and  3  the  lines  are  drawn  at  the  same  distances 
apart  as  in  the  corresponding  numbers  of  Lesson  I.  In  No.  4, 
three  lines  are  drawn  equidistant  between  the  ruled  lines. 
No.  2  represents  coarse  shading  ;  No.  3,  ordinary  shading ; 
and  No.  4,  fine  shading. 

Free-hand  exercises  on  the  blackboard,  similar  to  those  di- 
rected for  Lesson  I. 

LESSON  III.  Angles,  and  Plane  Figures.  —  No.  1  repre- 
sents two  right  angles,  x,  x,  formed  by  one  line  meeting  an- 
other. An  angle  is  the  opening  between  two  lines  that  meet. 

When  one  straight  line  (a  b)  falls  upon  another  straight 
line  (c  d),  so  as  to  make  the  adjacent  angles  (x,  x)  equal, 
the  two  angles  thus  formed  are  right  angles.  The  angle  at 
a*,  No.  2,  is  also  a  right  angle. 

An  acute  angle  (e)  is  an  angle  that  is  less  than  a  right 
angle ;  an  obtuse  angle  (n)  is  an  angle  that  is  greater  than 
a  right  angle. 

A  plane  is  a  surface,  on  which,  if  any  two  points  be  taken, 
the  straight  line  which  joins  them  touches  the  surface  in  its 
whole  length. 

Nos.  3,  4,  and  5  tare  plane  figures  called  squares. 

A  rectilinear  plane  figure  is  a  plane  figure  bounded  by 
straight  lines. 

A  square  is  a  plane  figure  that  has  four  equal  sides  and 


STRAIGHT   LINES    AXD   PLANE    SURFACES.  53 

four  right  angles.  Nos.  3, 4,  and  5  are  squares.  They  are 
also  called  erect  squares,  because  two  of  the  sides  of  each 
are  erect,  or  vertical. 

A  rectangle  is  a  four-sided  figure  having  only  right  an- 
gles. The  term  is  generally  applied  to  those  rectangular 
(right-angled)  figures  which  are  not  squares.  Nos.  6,  7,  and 
8  are  rectangles.  Nos.  9  and  10  may  be  divided  into  rect- 
angles. 

Principles  of  Surface  Measurement. 

We  will  suppose  that  throughout  Drawing-Book  No.  I. 
the  direct  distance  from  one  line  to  another  on  the  ruled 
paper  is  one  inch,  unless  otherwise  directed. 

Then,  how  much  space  will  one  of  the  small  ruled  squares 
contain  ?  (One  square  inch.)  How  much  will  four  of  them 
contain  ?  (Four  square  inches.)  As  a  standard  of  meas- 
urement, each  of  the  small  squares  formed  by  the  ruling  of 
the  paper  is  called  &  primary  erect  square. 

How  large  is  No.  3  ?  (One  inch  square.)  How  much 
area,  or  surface,  does  it  contain?  (One  square  inch.) 

How  large  is  No.  4  ?  (Two  inches  square.  That  is,  it 
measures  two  inches  on  each  side.)  How  much  area,  or  sur- 
face, does  it  contain?  (Four  square  inches,  as  may  be  seen 
by  counting  the  primary  squares  within  it.) 

How  large  is  No.  5  ?  (Four  inches  square.)  How  much 
area,  or  surface,  does  it  contain  ?  (Sixteen  square  inches.) 

How  large  is  No.  C  ?  (Two  inches  by  three  inches.)  How 
much  area,  or  surface,  does  it  contain  ?  (Six  square  inches.) 

How  large  is  No.  Y,  and  what  is  its  area?  How  large  is 
No.  8,  and  what  is  its  area  ? 

Hence, 

To  find  the  area  or  surface  measurement  of  any  rectangle : 

RULE  I. — Multiply  the  length  by  the  breadth,  and  the  prod- 
uct will  be  the  area. 

PROBLEMS  FOR  PRACTICE. 

1.  Draw  a  square  of  three  inches  to  a  side.     "What  is  its  area? 

Ans.  9  square  inches. 

2.  Draw  a  square  of  nine  inches  to  a  side.     "What  is  its  area  ? 

Ans.  81  square  inches. 


54  INDUSTRIAL   DRAWING.  [BOOK   NO.  I. 

3.  Draw  a  rectangle  of  four  by  five  inches.     What  is  its  area  ? 

Ana.  20  square  inches. 

4.  Draw  a  rectangle  of  six  by  eight  inches.     What  is  its  area  ? 

Ans. 

Let  the  Pupil  draw  the  foregoing  Problems  on  the  Hlac7c- 
board. 

No.  4  has  twice  the  length  of  sides  of  No.  3.  How  many 
times  larger  than  No.  3  is  it  ?  (Four  times  larger ;  because 
No.  3  contains  one  square  inch,  and  No.  4  contains  four 
square  inches.) 

No.  5  has  four  times  the  length  of  sides  of  No.  3.  How 
much  larger  than  No.  3  is  it  ?  (Sixteen  times  larger.) 

No.  5  has  twice  the  length  of  sides  of  No.  4.  How  much 
larger  than  No.  4  is  it  ?  (Four  times  larger.) 

No.  7  has  twice  the  length  of  sides  of  No.  G.  How  much 
larger  is  No.  7  than  No.  6  ?  (Four  times  larger.) 

From  the  foregoing  it  appears  that,  by  increasing  the 
lengths  of  the  sides  of  a  square  or  a  rectangle  to  two  times 
their  length,  we  form  a  similar  figure  four  times  as  large ; 
by  increasing  to  three  times,  we  form  a  similar  figure  nine 
times  as  large ;  by  increasing  to  four  times,  we  form  one 
sixteen  times  as  large  ;  by  increasing  to  Jive  times,  we  form 
one  twenty -five  times  as  large,  etc.  The  same  princi- 
ple holds  true  with  regard  to  a  figure  of  any  number  of 
sides. 

ELEMENTARY  PRINCIPLE. —  The  areas  of 'similar  plane  fig- 
ures are  as  the  squares  of  their  similar  sides. 

If,  therefore,  we  have  a  plane  figure  of  any  number  of 
sides,  and  wish  to  make  another  similar  to  it,  lout  four  times 
as  large,  we  double  the  lengths  of  the  sides ;  because  2 
times  2  are  four :  if  we  wish  to  make  one  nine  times  as 
large,  we  treble  the  lengths  of  the  sides ;  because  3  times  3 
arc  nine:  if  we  wish  to  make  one  sixteen  times  as  large,  we 
quadrille  the  lengths  of  the  sides ;  because  4  times  4  are 
sixteen:  and  so  on  to  the  square  of  any  given  number. 

IrIP  Let  the  teacher  explain  more  fully,  if  necessary, 
what  is  meant  by  the  square  of  a  number,  and  especially 
when  that  number  represents  the  length  of  a  given  line. 


STRAIGHT   LINES   AND    PLANE   SURFACES.  55 

PROBLEMS   FOR   PRACTICE. 

1.  Draw  a  square  similar  to  No.  3,  but  nine  times  as  large. 

2.  Draw  a  square  similar  to  No.  4,  but  nine  times  as  large. 

3.  Draw  a  square  similar  to  No.  4,  but  twenty-five  times  as  large. 

4.  Draw  a  rectangle  similar  to  No.  6,  but  nine  times  as  large. 

5.  Draw  a  rectangle  similar  to  No.  6,  but  four  times  as  large. 
G.  Draw  a  polygon  similar  to  No.  10,  but  four  times  as  large. 

A  polygon  is  a  plane  figure  having  many  sides  and  miiny 
angles.  The  term  is  generally  applied  to  a  plane  figure  of 
more  than  four  angles  and  four  sides. 

Free-hand  exercises  on  the  blackboard. — Let  the  pupil  fol- 
low out,  on  the  blackboard,  a  course  of  exercises  similar  to 
those  prescribed  for  the  tint-lined  drawing-paper. 

LESSON  IV.  Diagonals. — Diagonals  are  lines  drawn  in 
the  direction  of  a  diagonal  of  a  primary  erect  square. 

A  primary  diagonal  is  a  line  drawn  diagonally  from  one 
corner  to  another  of  a  primary  erect  square. 

No.  1  is  made  up  of  primary  diagonals  in  two  directions. 

No.  2  is  a  primary  diagonal  square.  What  is  its  area 
equal  to  ?  (Two  square  inches ;  inasmuch  as  it  includes 
four  halves  of  the  small  primary  erect  squares.) 

What  is  the  area  of  No.  3  ?* 

What  is  the  area  of  No.  4  ?    No.  5  ?     No.  6  ? 

If  No.  2  have  its  sides  doubled  in  length,  how  much  larger 
will  the  figure  be  ? 

If  No.  2  have  its  sides  trebled  in  length,  how  much  larger 
will  the  figure  be  ? 

PROBLEMS   FOR   PRACTICE. 

1.  Draw  a  diagonal  square  similar  to  No.  2,  but  sixteen  times  as  large; 
that  is,  containing  sixteen  times  the  area  of  No.  2.     How  long  must  the 
sides  be,  compared  with  the  sides  of  No.  2  ? 

2.  Draw  a  diagonal  square  similar  to  No.  2,  but  twenty-five  times  as 
large.     How  long  must  the  sides  be,  compared  with  the  sides  of  No.  2  ? 

3.  Draw  a  diagonal  square  similar  to  No.  3,  but  nine  times  as  large. 

4.  Draw  a  diagonal  rectangle  similar  to  No.  5,  but  four  times  as  large. 

*  The  halves  of  square  inches  included  within  the  figures  in  this  lesson 
might  be  marked  with  dots,  for  greater  facility  in  counting  them. 


56  INDUSTRIAL   DRAWING.  [BOOK   NO.  1. 


Iii  drawing  these  problems  let  the  pupils  arrange 
them  in  such  a  manner  as  to  economize  the  space  on  the 
drawing-paper. 

To  find  the  area  of  any  diagonal  square,  or  other  diagonal  rectangle : 

RULE  A. — Multiply  the  length  in  primary  diagonals  by 
the  breadth  in  primary  diagonals,  and  TWICE  the  product 
will  be  the  area,  in  measures  of  the  primary  erect  squares. 

Kule  A  is  only  a  special  application  of  Rule  I. 

(Reason  for  the  ride. — The  length  in  primary  diagonals 
multiplied  by  the  breadth  in  primary  diagonals  will  give 
the  number  of  primary  diagonal  squares ;  and  we  then  mul- 
tiply by  2,  because  there  are  two  primary  erect  squares  in 
each  primary  diagonal  square.) 

Thus,  in  No.  3,  multiply  2,  the  length  in  primary  diago- 
nals of  one  side,  by  2,  the  length  in  primary  diagonals  of 
another  side,  and  the  product  will  be  4 ;  and  twice  four  will 
be  the  area  in  primary  erect  squares,  or  square  inches. 

What  is  the  area  of  a  diagonal  square  of  7  diagonals  to 
a  side  ?  (Ans.  98  square  inches.) 

What  is  the  area  of  a  diagonal  rectangle  of  5  by  7  diag- 
onals ?  (Ans.  70  square  inches.) 

Let  the  pupil  carry  out  the  same  system  on  the  black- 
board. 

LESSON  Y. — No.  1  is  an  erect  cross,  representing  one  thin 
piece,  2  inches  by  8  inches,  laid  at  right  angles  across 
another  piece  2  inches  by  6  inches.  First  draw  the  upper 
piece,  marked  1,  and  shade  it  lightly.  The  lower  piece 
might  have  the  shading  described  in  No.  4  of  Lesson  II. 

No.  2.  Draw  the  pieces  in  the  order  in  which  they  are 
numbered.  The  lower  piece  is  first  shaded  with  diagonal 
lines,  the  same  as  the  upper  piece,  and  the  shading  is  finish- 
ed by  drawing  lines  between  the  diagonals  first  drawn. 

Nos.  3  and  4.  In  these,  and  in  all  similar  figures,  the  up- 
per pieces — supposing  that  the  pieces  are  in  a  horizontal 
position — should  be  drawn  first.  In  most  outline  draw- 
ings, and  in  lightly  shaded  drawings,  the  outline  is  made 
heaviest  on  the  side  opposite  to  the  direction  from  which 


STRAIGHT    LINES    AND   PLANE    SURFACES.  57 

the  light  is  supposed  to  come.  Thus,  in  No.  4,  the  light 
is  supposed  to  come  in  the  direction  of  the  arrow  «/  and 
hence  the  outlines  are  made  the  heaviest  where  the  shad- 
ows would  naturally  fall. 

No.  5.  Observe  the  direction  in  which  the  light  falls  upon 
this  figure,  as  indicated  by  the  arrow  #,  and  the  consequent 
heavy  outlines  of  those  sides  of  the  four  pieces  which  would 
be  in  shadow. 

The  shading  in  No.  7  should  render  each  square  distinct 
from  the  others. 

No.  8  is  a  pattern  made  up  of  only  one  figure,  repeated 
continuously,  and  so  arranged  as  to  cover  the  entire  sur- 
face. A  very  great  variety  of  patterns,  consisting  wholly 
of  repetitions  of  one  figure  to  each  pattern,  may  easily  be 
designed,  and  drawn  by  the  aid  of  the  ruled  paper. 

What  is  the  area  of  each  of  the  squares,  as  they  are  num- 
bered, in  No.  7. 

The  area  of  the  pattern  figure  in  No.  8  ? 

Free-hand  exercises  on  the  blackboard. 

PAGE  TWO. 

LESSON  VI.  Tico-space  Diagonals. — By  a  two-space  diag- 
onal is  meant  the  diagonal  of  a  rectangle  which  is  twice  as 
long  as  it  is  broad.  It  is  a  diagonal  which  passes  over  two 
spaces  on  the  ruled  paper. 

No.  1.  The  lines  in  No.  1  are  two-space  diagonals.  They 
should  be  copied,  without  the  aid  of  a  ruler,  until  they  can 
be  drawn  with  tolerable  accuracy,  and  with  facility.  At  b 
lines  are  first  drawn  as  at  a;  and  then  lines  are  drawn  in- 
termediate between  them;  c  is  first  drawn  the  same  as  bt 
and  is  then  filled  in  with  intermediates.  In  this  manner 
great  uniformity  of  shading  may  be  attained. 

No.  2  is  drawn  in  a  manner  similar  to  No.  1.  First  trace 
each  line  lightly,  and  continue  to  pass  the  pencil  over  it  un- 
til it  is  drawn  with  accuracy. 

No.  3.  As  two  square  inches  are  represented  in  the  dot- 
ted rectangle,  and  as  the  line  a  b  divides  the  rectangle  into 
two  equal  parts,  therefore  on  each  side  of  the  line  there  is 
an  area  equal  to  one  square  inch, 

O  9 

~     "  "         THE 


58  INDUSTRIAL   DEAWIXG.  [BOOK   NO.  I. 

No.  4.  What  area  is  embraced  within  the  dotted  square  ? 
Then  how  much  is  embraced  within  the  portion  a  f 

No.  5.  What  area  is  embraced  within  the  dotted  rectan- 
gle ?  Then  what  area  is  embraced  within  the  portion  a  ? 

The  portion  marked  a  in  No.  4  is  a  triangle — a  figure  of  three  sides  and 
three  angles.  It  is  an  acute-angled  triangle,  because  each  angle  is  less  than 
a  right  angle.  (See  Lesson  III.)  The  portion  marked  a  in  No.  5  is  called 
an  obt use-angled  triangle,  because  one  of  the  angles  is  greater  than  a  right 
angle. 

No.  6  is  a  figure  called  a  rhombus.  A  rhombus  is  a  figure 
which  has  four  equal  sides,  the  opposite  sides  being  parallel ; 
but  its  angles  are  not  right  angles.  What  area  is  embraced 
in  the  upper  half  of  No.  6  ?  In  the  whole  figure  ? 

No.  Y.  What  area  is  embraced  in  each  of  the  parts  a  of 
No.  7  ?  In  the  central  rectangle  b  f  In  the  whole  rhom- 
bus? (16  square  inches.) 

No.  8.  In  the  dotted  figure  No.  8  there  are  three  of  the 
small  squares ;  hence  the  dotted  figure  contains  an  area  of 
three  square  inches.  But  the  part  b  (as  shown  in  No.  3  and 
No.  5)  contains  an  area  of  one  square  inch,  and  the  part  c  an 
area  of  one  square  inch ;  hence  the  part  a  must  contain  an 
area  of  one  square  inch  also. 

No.  9.  What  area  is  embraced  in  the  rhombus  No.  9? 
(Let  the  pupil  prove  that  each  part  a  embraces  an  area  of 
one  square  inch,  the  same  as  a  in  No.  8.) 

No.  10.  What  area  is  embraced  in  No.  10?  How  is  it 
shown  that  the  upper  part  marked  1  contains  an  area  equal 
to  one  square  inch  ? 

No.  11.  What  area  is  embraced  in  the  star  figure  No.  11  ? 
(Let  the  pupil  prove  that  each  of  the  points  marked  1  con- 
tains an  area  of  one  square  inch.) 

No.  12  is  an  octagonal  or  eight-sided  figure.  A  regular 
octagon  has  eight  equal  sides  and  eight  equal  angles;  but 
here,  while  the  sides  are  equal,  the  angles  are  not  all  equal. 
What  is  the  area  of  each  of  the  parts  a  of  the  octagon  ? 
Of  the  whole  octagon  ? 

The  shading  of  the  central  square  of  No.  12  is  produced  by  carrying  the 
pencil  from  left  to  right  with  a  running  dotting  motion.  In  industrial 
drawing  it  is  desirable  to  designate  the  different  sides  or  surfaces  of  objects 


STRAIGHT  LINES   AND   PLANE    SURFACES.  59 

very  distinctly  by  the  shading ;  and  this  is  one  of  the  kinds  of  shading 
very  appropriate  for  that  purpose. 

No.  13.  What  is  the  area  of  each  of  the  rhombuses  mark- 
ed a  f  (See  No.  4  and  No.  6.)  What  is  the  area  of  the 
central  star  figure  ?  (See  No.  11.)  What  is  the  area  of  the 
whole  octagon  ? 

No.  14.  What  is  the  area  of  each  of  the  rhombuses  marked 
a  ?  (See  No.  9.)  Of  each  of  the  star  points  marked  b  f  (See 
No.  4.)  Of  the  central  square  c  f  Of  the  whole  octagon  ? 

PROBLEMS   FOB  PRACTICE. 

1.  Draw  a  rectangle  similar  to  the  dotted  rectangle  No.  4.  but  four  times 
as  large.     (See  ELEMENTARY  PRINCIPLE,  page  54.) 

2.  Draw  a  triangle  similar  to  a  of  No.  4,  but  nine  times  as  large.     How 
must  the  sides  compare  in  length  with  those  of  a  of  No.  4  ?    What  will 
be  the  area  of  the  triangle  ? 

3.  Draw  a  triangle  similar  to  a  of  No.  5,  but  containing  sixteen  times  the 
area  of  No.  5. 

4.  Draw  a  rhombus  similar  to  No.  6,  but  containing  twenty-five  times  the 
area  of  No.  6.     Shade  it  with  two-space  diagonals  like  6,  or  c,  of  No.  1 . 

5.  Draw  a  rhombus  similar  to  No.  7,  but  containing  only  one  fourth  the 
area  of  No.  7. 

6.  Draw  a  figure  similar  to  the  a  portion  of  No.  8,  but  sixteen  times  as 
large. 

7.  Draw  a  rhombus  similar  to  No.  9,  but  containing  twenty-five  times  the 
area  of  No.  9. 

8.  Draw  a  rhombus  similar  to  No.  10,  but  having  four  times  the  area  of 
No.  10. 

9.  Draw  a  star  figure  similar  to  No.  11,  but  containing  nine  times  the  area 
of  No.  11. 

10.  Draw  an  octagon  similar  to  No.  12,  but  containing  four  times  the  area 
of  No.  12. 

11.  Draw  an  octagon  similar  to  No.  13,  but  containing  nine  times  the  area 
of  No.  13.     Divide  it  as  No.  1 3  is  divided,  and  mark  within  each  rhombus, 
its  area,  and  mark  the  area  of  the  star  also. 

12.  Draw  a  figure  similar  to  No.  14,  but  sixteen  times  as  large,  and  mark 
within  the  parts  a,  6,  and  c  the  area  of  each. 

Let  problems  similar  to  the  foregoing  be  drawn  on  the 
blackboard,  or  selections  from  them,  at  the  option  of  the 
teacher. 

LESSON  VII. — No.  1  is  a  two-space  diagonal  square ;  and 
No.  2  is  the  same  in  a  different  position.  The  area  of  No. 
1  can  easily  be  counted  up,  when  it  is  seen  that  each  of  the 


60  INDUSTRIAL    DRAWING.  [BOOK   NO.  I. 

parts  marked  1  is  equal  to  one  square  inch.  Hence  the  fig- 
ure contains  five  square  inches. 

The  area  of  a  two-space  diagonal  square,  or  of  any  two- 
space  diagonal  rectangle,  may  be  found  by  the  following 
modification  of  Rule  I. : 

To  find  the  area  of  a  two-space  diagonal  rectangle : 

RULE  B. — Multiply  the  length  in  two-space  diagonals  by 
the  breadth  in  two-space  diagonals,  and  FIVE  times  the  prod- 
uct  will  be  the  area,  in  measures  of  the  primary  erect  squares. 

Thus,  in  No.  1,  multiply  the  length  1  by  the  breadth  1,  and 
5  times  the  product  will  be  the  area:  5  square  inches. 

What  is  the  area  of  the  two-space  diagonal  square  No.  3  ? 

Solution. — Multiply  the  length  2  by  the  breadth  2,  and 
the  product  will  be  4,  which,  multiplied  by  5,  will  give  20 
square  inches — the  area.  The  same  result  will  be  found  by 
counting  the  squares,  etc. 

What  is  the  area  of  No.  4  ?     No.  5  ?     No.  6  ? 

No.  7  is  the  same  form  of  star  seen  in  No.  14  of  Lesson 
VI. ;  and  No.  8  is  the  same  form  that  is  seen  in  No.  13.  In 
drawing  these  figures,  first  trace  the  outlines  very  lightly; 
and  do  not  mark  firmly  until  the  positions  of  all  the  lines 
are  clearly  determined.  Use  no  ruler. 

No.  9  shows  two  octagons  intersecting  each  other  in  a 
diagonal  direction,  and  in  such  a  manner  that  the  rhombus 
a  is  common  to  both.  Any  octagon  may  have  an  octagon 
intersecting  it  in  this  manner  on  all  of  its  four  divisions; 
and  when  the  series  is  continued  they  form  the  pattern 
seen  in  No.  10 — sometimes  seen  in  oil-cloths,  carpets,  etc. 

No.  1 1  shows  a  series  of  octagons  intersecting  one  another 
vertically  and  horizontally,  instead  of  diagonally  as  in  No. 
10.  In  No.  10  the  rhombuses,  and  in  No.  11  the  star  fig- 
ures, are  represented  as  shaded  with  a  light  tint  of  India  ink. 

PROBLEMS   FOR   PRACTICE. 

1.  Draw,  on  the  drawing-paper,  a  two-space  diagonal  square,  similar  to 
No.  1,  but  embracing  twenty-five  times  the  area  of  No.  1.  Draw  another 
within  the  last,  embracing  nine  times  the  area  of  No.  1.  What  will  be  the 
lengths  of  the  sides  of  each,  in  two-space  diagonals  ?  The  area  of  the  small- 
er square  ?  Of  that  portion  of  the  larger  square  outside  of  the  smaller  ? 


STRAIGHT   LINES  AND   PLANE    SURFACES.  61 

2.  Draw  a  rectangle  similar  to  No.  4,  but  embracing  four  times  the  area 
of  No.  4. 

3.  Draw  a  rectangle  similar  to  No.  6,  but  embracing  four  times  the  area 
of  No.  6. 

4.  What  area  would  be  included  in  a  two-space  diagonal  rectangle  hav- 
ing a  length  of  eight  two-space  diagonal  measures,  and  a  breadth  of  five  ? 
(See  Rule  B.) 

5.  Draw  a  pattern  similar  to  No.  10,  but  with  the  figures  embracing  four 
times  the  area  of  those  in  No.  10. 

6.  Draw  a  pattern  similar  to  No.  11,  but  with  the  figures  embracing  four 
times  the  area  of  those  in  No.  11. 

Free-hand  drawing  of  problems  similar  to  the  foregoing 
on  the  blackboard. 

LESSON  YIIL  Three-space  Diagonals. — By  a  three-space 
diagonal  is  meant  the  diagonal  of  a  rectangle  which  is  three 
times  as  long  as  it  is  broad.  Thus,  the  diagonal  of  the  rect- 
angle at  cr,  No.  1,  passes  over  three  spaces,  and  divides  the 
rectangle  into  two  equal  parts.  As  the  rectangle  includes 
an  area  of  three  square  inches,  each  half  of  it,  as  marked  ^, 
has  an  area  of  one  and  a  half  square  inches. 

No.  2.  What  area  is  included  in  the  dotted  rectangle  No. 
2  ?  In  each  of  the  three  parts,  a,  b,  and  c  ? 

No.  3  is  a  three -space  diagonal  square.  Observe  that 
each  of  the  parts  marked  1  has  an  area  of  one  and  a  half 
square  inches.  Then  what  is  the  area  of  the  whole  square  ? 
(Ans.  10  square  inches.) 

No.  4  is  a  rhombus.  What  is  its  area  ?  The  area  of  the 
dotted  rectangle  ? 

No.  5  is  a  three -space  diagonal  rectangle.  Its  area  is 
easily  found,  by  counting,  to  be  twenty  square  inches.  But 
the  area  of  any  three-space  diagonal  square,  or  other  three- 
space  diagonal  rectangle,  however  large,  may  easily  be  found 
by  the  following  rule,  also  a  modification  of  Rule  I. 

To  find  the  area  of  a  three-space  diagonal  rectangle : 

RULE  C. — Multiply  the  length  in  three-space  diagonals  by 
the  breadth  in  three-space  diagonals,  and  TEN  times  the  prod- 
uct will  be  the  area,  in  measures  of  the  primary  erect  squares. 

Thus,  in  the  square  No.  3,  the  length  1,  of  one  side,  multi- 
plied by  1,  the  length  of  another  side,  gives  the  product 


G2  INDUSTRIAL   DRAWING.  [BOOK    NO.  I. 

1,  which,  multiplied  by  10,  gives  10  square  inches  as  the 
area. 

Apply  the  rule  to  No.  5,  and  test  the  result  by  counting. 
What  is  the  area  ? 

No.  6.  What  is  the  area  of  the  inner  dotted  rectangle? 
Of  the  large  rectangle  ?  Then  what  is  the  area  of  the  space 
included  between  the  two  ? 

"No.  7.  The  area  of  the  space  included  within  the  dotted 
figures,,./,  29  3  is  seen,  by  counting,  to  be  five  square  inches. 
But  the  area  of  the  part  marked  a  is  one  and  a  half  square 
inches,  and  the  area  of  c  is  the  same,  the  two  parts  a  and  c 
making  three  square  inches.  Therefore  the  part  b  embraces 
two  square  inches. 

No.  8.  What  is  the  area  of  the  rhombus  No.  8  ? 

No.  9  is  a  three -space  diagonal  octagon.  What  is  the 
area  of  each  of  the  parts  «,  #,  c,  and  df  Of  the  inner  dotted 
square  ?  Of  the  whole  octagon  ? 

No.  10.  What  is  the  area  of  the  four  rhombuses  a,  #,  c,  d? 
Of  the  star  g  ?  Of  the  whole  octagon  ? 

No.  11.  What  is  the  area  of  the  four  rhombuses  a,  £,  c,  d? 
Of  the  four  parts  e,  /,  g,  h  ?  Of  the  central  square  k  f  Of 
the  whole  octagon  ? 

No.  12.  What  is  the  area  of  the  star  in  No.  12  ?  What  is 
the  area  of  the  star  in  No.  8  of  Lesson  VII.  ?  What  is  the 
difference  in  their  areas  ? 

Let  all  the  foregoing  be  drawn  on  the  drawing-paper. 

PROBLEMS   FOR   PRACTICE. 

1 .  Draw,  on  the  drawing-paper,  a  three-space  diagonal  square  that  shall 
contain  9  times  the  area  of  No.  3. 

2.  Draw  a  rhombus  similar  to  No.  4,  but  containing  nine  times  the  area 
of  No.  4.     Within  the  rhombus  thus  drawn,  and  equidistant  from  its  sides, 
draw  a  rhombus  containing  four  times  the  area  of  No.  4.    Within  this  latter, 
and  equidistant  from  its  sides,  draw  another  equal  to  No.  4.     Mark  the 
rhombuses  thus  drawn  No.  1,  No.  2,  and  No.  3,  beginning  with  the  smallest, 
and  mark  the  area  of  each. 

3.  Draw  a  rectangle  similar  to  No.  5,  but  containing  sixteen  times  the  area 
of  No.  5.    Draw  one  within  this  latter  containing  four  times  the  area  of  No.  5. 

4.  Draw  a  rhombus  similar  to  No.  8,  but  containing  four  times  the  area 
of  No.  8. 

5.  Draw  an  octagon  similar  to  No.  9,  but  containing  four  times  the  area 
of  No.  9. 


STRAIGHT   LINES   AND   PLANE    SURFACES.  63 

6.  Draw  an  octagon  similar  to  No.  10,  but  with  other  interlacing  octa- 
gons on  its  diagonal  sides,  similar  to  No.  10  of  Lesson  VII. 

7.  Draw  an  octagon  similar  to  No.  11,  but  with  other  interlacing  octa- 
gons on  its  vertical  and  horizontal  sides,  similar  to  No.  11  of  Lesson  VII. 

8.  Draw  a  star  similar  to  g  of  No.  10,  but  having  four  times  the  area  of 
ff,  and  inclose  it  with  an  interlacing  square  similar  to  No.  7  of  Lesson  VII. 

Free-hand  drawing  of  problems  similar  to  the  foregoing 
on  the  blackboard. 

PAGE  THREE. 

LESSON  IX. — This  lesson  consists  of  a  series  of  net-work, 
the  finer  examples  of  which,  when  used  in  drawing  or  en- 
graving, for  the  purposes  of  shading,  are  called  hatching. 

No.  1  is  a  coarse  diagonal  net-work,  in  the  form  of  squares. 

No.  2  is  drawn,  in  the  first  place,  in  the  same  manner  as 
No.  1 ;  after  which  another  set  of  lines  is  put  in,  in  both 
diagonal  directions,  intermediate  between  those  first  drawn. 

No.  3  is  first  drawn  the  same  as  No.  2,  after  which  another 
set  of  lines  is  put  in  intermediate  between  those  first  drawn. 
This  kind  of  hatching  is  seen  in  No.  6  of  the  next  lesson. 

No.  4  is  a  coarse  two-space  diagonal  net-work. 

No.  5  is  first  drawn  the  same  as  No.  4,  and  is  then  filled 
in  with  another  set  of  lines  between  those  first  drawn. 

No.  6  is  a  fine  hatching,  first  drawn  the  same  as  No.  5,  and 
then  filled  in  with  another  set  of  lines  intermediate  between 
those  first  drawn.  A  sharp-pointed,  hard  pencil  is  required 
for  this  shading. 

No.  7  is  a  coarse  three-space  diagonal  net-work.  When 
filled  in  with  two  lines  intermediate  between  those  here 
drawn,  it  forms  a  good  hatching  for  some  kinds  of  shading. 

All  the  examples  in  this  lesson,  which  should  be  copied 
without  the  aid  of  a  ruler,  will  furnish  good  exercises  in 
drawing  straight,  uniform,  and  equidistant  lines.  The  di- 
rections, and  the  distances  apart,  are  given  in  the  ruling  of 
the  paper. 

Free-hand  drawing  of  Lesson  IX.  on  the  blackboard. 

LESSON  X. — No.  1  gives  the  outline  of  a  star-shaped  fig- 
ure ;  and  No.  2  is  the  same  divided  into  eight  pairs  of  wings 
by  a  vertical,  a  horizontal,  and  two  diagonal  lines,  and  then 


64  INDUSTRIAL   DRAWING.  [BOOK    NO.  T. 

shaded.  This  peculiar  star-shaped  figure  is  a  common  form 
of  ornament  in  examples  of  Byzantine  art.  What  is  the 
area  of  each  of  the  eight  pairs  of  wings  of  No.  2  ?  Of  the 
whole  star? 

No.  3  is  a  star  similar  to  No.  2,  inclosed  in  a  diagonal 
square,  but  with  twice  the  length  of  sides  of  No.  2.  How, 
then,  does  its  area  compare  with  that  of  No.  2  ?  What  is 
the  area  of  the  diagonal  square?  (See  Rule  A,  page  56.) 

No.  4  is  a  hexagonal  pattern  covering  the  entire  surface. 
A  hexagon  is  a  plane  figure  of  six  sides  and  six  angles. 
When  the  sides  are  all  equal,  and  the  angles  all  equal,  it  is 
a  regular  hexagon.  What  is  the  area  of  one  of  the  hexagons 
of  No.  4? 

No.  5  is  a  pattern  composed  of  an  elongated  octagonal 
figure  and  a  square,  the  two  forms  combined  covering  the 
whole  surface.  What  is  the  area  of  one  of  the  octagons  ? 

Nos.  4  and  5  may  be  varied  so  as  to  embrace  a  great  va- 
riety of  similar  patterns  by  changing  the  relative  lengths 
of  the  sides.  Numerous  oil-cloth  and  carpet  patterns  are 
formed  on  this  basis.  Additional  variety  is  given  to  Nos. 
4  and  5  by  the  bordering,  as  indicated  at  a  and  b.  Observe 
that  the  exact  distance  of  the  inner  lines  from  the  outer 
border  is  given  by  the  intersections  of  the  ruled  lines. 

No.  6  is  an  original  Moorish  pavement  pattern,  called  mo- 
saic /  but  it  is  now  common,  with  various  modifications,  in 
pavements,  oil-cloths,  etc.  It  is  easily  drawn  on  the  ruled 
paper.  The  hatching  used  in  the  shading  is  that  of  No.  3 
of  Lesson  IX. 

No.  7  is  an  elongated  hexagonal  link  pattern,  for  borders, 
etc.  Observe  the  position  of  the  heavy  shaded  lines  on  the 
right  hand  and  lower  sides. 

No.  8  is  a  double  interlacing  square.     First  trace  lightly. 


PROBLEMS  FOR 

1.  Draw  a  star  similar  to  No.  2,  but  nine  tittles  as  lai'ge.     What  will  le 
its  area  ? 

2.  Draw  a  star  similar  to  No.  3,  but  four  times  as  large,  and  inclose  it 
with  a  diagonal  square  similar  to  the  inclosure  of  No.  3.     Centrally  within 
each  of  the  corner  diagonal  squares  similar  to  o,  6,  c,  d  of  No.  3  place  a  star 
Ufce  No,  2  within  its  qwi]  diagqnaj  square,    '  ......  \ 


STRAIGHT  LINES   AND   PLANE    SURFACES.  65 

3.  Draw  a  pattern  made  up  of  hexagons  similar  to  those  of  Xo.  4,  but  of 
four  times  the  area,  and  complete  each  in  a  manner  similar  to  a. 

4.  Draw  a  pattern  like  No.  5,  but  with  the  exception  that  the  shade;1, 
squares  shall  contain  four  times  the  area  of  those  in  No.  5.     Complete  the 
several  hexagons  in  a  manner  similar  to  b. 

5.  Draw  a  pattern  composed  wholly  of  figures  like  No.  G.     The  addition 
at  a  will  show  how  the  several  figures  are  to  be  connected.     Shade  all  like 
No.  6j  or  use  different  tints  of  India  ink. 

G.  Draw  a  link  pattern  like  No.  7,  with  the  exception  that  each  link  shall 
be  two  spaces  longer  than  in  No.  7,  but  of  the  same  width. 

free-hand  drawing  of  Lesson  J£,  and  the  problems,  on 
the  blackboard. 

LESSON  XL — No.  1  and  No.  2  are  the  elements  of  slightly 
different  patterns  formed  on  the  basis  of  either  an  erect  or 
a  diagonal  square. 

No.  3  is  the  basis  of  an  octagonal  pattern.  At  No.  4  the 
short  lines  a  a  a  a  show  the  method  of  marking  out  an  in- 
ner octagon  whose  sides  shall  be  uniformly  distant  from  and 
parallel  to  the  sides  of  the  larger  octagon  ;  and  No.  5  shows 
the  figure  completed. 

No.  6  is  the  pattern,  as  carried  out,  from  the  preceding 
three  figures.  The  central  octagons  should  be  shaded  with 
a  light  tint  of  India  ink,  and  the  squares  with  the  running 
dots  in  horizontal  lines.  The  different  kinds  of  shading 
used  in  the  patterns  given  in  these  books  denote  the  variety 
of  colors  employed  when  the  pattern  is  used  either  in  orna- 
mental art,  or  in  the  designs  of  oil-cloths,  carpets,  wall-pa- 
per, etc. 

No.  7  is  a  dodecagon — a  figure  of  twelve  sides  and  twelve 
angles.  When  the  sides  are  equal,  and  the  angles  equal, 
the  figure  is  a  regular  dodecagon.  What  is  the  area  of 
each  of  the  parts  a,  £,  c,  d  of  this  figure  ?  Of  the  central 
dotted  square  ?  Of  the  whole  dodecagon  ? 

No.  8  is  a  dodecagon  divided  into  a  border  of  squares  and 
triangles,  and  a  central  hexagon.  What  is  the  area  of  each 
of  the  two  vertical  squares  ?  Of  each  of  the  four  two-space 
diagonal  squares?  Of  each  of  the  six  white  triangles  that 
incloses  the  small  dark  triangle?  Of  the  central  hexagon.9 
Of  the  whole  figure  ? 


66  INDUSTRIAL   DRAWING.  [BOOK   NO.  I. 

No.  9  is  a  pattern  composed  wholly  of  intersecting  dodec- 
agons like  No.  8.  Each  figure,  it  will  be  seen,  forms  a  por- 
tion .of  six  other  like  figures  surrounding  it.  This  combina- 
tion of  dodecagons  is  an  original  pavement  pattern  taken 
from  a  Roman  church  in  the  Byzantine  period  of  Roman 
history.  It  is  an  admirable  specimen  of  geometrical  mosaic 
work  so  common  in  that  period;  and  it  must  have  been 
formed  upon  lines  drawn  precisely  like  those  given  on  our 
ruled  paper;  for  in  no  other  manner  could  a  series  of  such 
figures  be  drawn  with  accuracy. 

PROBLEMS    FOR   PRACTICE. 

1.  Draw  a  pattern  formed  of  figures  like  No.l,  allowing  the  figures  to 
touch  vertically  and  horizontally.     Give  to  these  figures  the  running  dot 
shading;  and  shade  the  intermediate  figures  formed  between  them  with  a 
light  tint  of  India  ink. 

2.  Draw  a  pattern  formed  of  figures  like  No.  2,  and  shade  the  diagonal 
cross  with  a  tint  of  India  ink,  and  the  intermediate  figures  with  the  running 
dot  shading.     Leave  the  diagonal  squares  unshaded. 

3.  Draw  a  double  interlacing  square  like  No.  8  of  Lesson  X.,  and  cen- 
trally within  it  draw  a  figure  like  No.  8  of  Lesson  XI. 

Patterns  similar  to  No.  6  and  No.  9  may  be  drawn  4,  9,  16,  25,  or  3G 
times,  etc.,  larger  than  the  figures  here  given,  by  increasing  the  lengths  of 
the  sides  2,  3,  4,  5,  or  G  times,  etc. ,  according  to  the  principles  explained 
on  pages  53  and  54. 

Free-hand  drawing  of  Lesson  XI.,  and  the  problems,  on 
the  blackboard. 


PAGE  FOUR. 

For  convenience,  we  now  drop  the  method  of  grouping 
the  examples  under  the  head  of  Lessons,  and  here  designate 
them  as  separate  Figures. 

Fig.  1  represents  an  ancient  Egyptian  pattern  of  a  braid- 
ed or  woven  mat  on  which  the  king  stood.  It  is  formed  of 
flat  strands  of  only  two  colors,  each  strand  passing  continu- 
ously, in  a  diagonal  direction,  over  two  strands  of  different 
colors,  and  then  under  two.  The  portion  of  a  strand  pre- 
sented at  one  view  is  rectangular,  and  twice  as  long  as  it  is 
broad.  All  the  lines  in  this  figure  are  diagonals,  and  should 
be  drawn  without  the  aid  of  a  ruler.  The  shaded  strands 


STRAIGHT   LINES   AND   PLANE    SURFACES.  67 

may  be  gone  over,  first  lightly,  with  India  ink,  and  then 
with  pencil. 

Fig.  2  is  another  Egyptian  pattern  of  matting,  in  only 
two  colors,  but  presenting  a  view  quite  different  from  Fig.  1. 
Here  each  light  strand  passes  continuously  over  two  dark 
strands,  and  then  under  three  dark  strands.  The  dark 
strands  may  be  considered  as  the  icarp,  and  are  arranged 
side  by  side,  all  running  diagonally;  and  then  the  light 
strands,  being  the  filling,  are  woven  in  diagonally,  as  stated, 
at  right  angles  to  the  warp.  Patterns  similar  to  Figs.  1  and 
2  may  be  formed  of  worsted  of  two  colors. 

Fig.  3  is  the  pattern  of  an  Arabian  pavement  found  at 
Cairo,  formed  of  black  and  of  white  marble,  except  the  diag- 
onal squares,  which  are  of  red  tile.  Go  over  the  diagonal 
squares  once,  and  the  rectangles  twice,  with  a  light  tint  of 
India  ink.  The  Arabians  imitated  the  universal  practice  of 
the  Komans  of  covering  the  floors  of  their  public  buildings, 
mosques,  etc.,  with  mosaic  patterns  arranged  on  a  geomet- 
rical system. 

Fig.  4  is  a  decorative  pattern,  in  different  colors,  from  an 
ancient  Egyptian  tomb.  It  is  supposed  to  have  suggested 
the  meander,  or  fret,  to  the  Greeks.  (See  page  6  of  draw- 
ings.) The  ruler  may  be  used  in  this  figure,  after  first  indi- 
cating the  lines  with  the  pencil  alone. 

Fig.  5  is  an  ancient  Egyptian  pattern,  in  different  col- 
ors, from  the  painting  on  a  tomb.  In  most  of  these  Egyp- 
tian paintings  the  colors  are  as  fresh  as  if  put  on  yesterday. 

Fig.  6  is  an  octagonal  pattern  forming  intermediate  fig- 
ures of  diagonal  squares.  The  ruled  lines  furnish  conven- 
ient guides  for  forming  the  width  of  the  octagon  border. 

Figs.  7  and  8  are  samples  of  mosaic  patterns  based  upon 
two  of  the  forms  of  the  central  eight-pointed  star  figure,  so 
common  in  specimens  of  Byzantine  ornamental  art. 

Fig.  9  is  another  modification  of  the  star  figure  in  mosaic, 
here  inclosed  by  an  interlacing  border. 

Fig.  10  represents  a  portion  of  a  mosaic  pavement,  in  dif- 
ferent colors,  from  the  ruins  of  Pompeii.  Observe  that  the 
running  dotted  shading  is  done  very  lightly,  and  with  a 
sharp  pencil,  in  Figs.  7,  8,  and  9  ;  but  much  more  heavily,  and 
with  a  blunt  pencil,  in  a  portion  of  Fig.  10. 


68  INDUSTRIAL   DRAWING.  [BOOK   NO.  I. 


PROBLEMS    FOR   PRACTICE. 

1.  Draw  a  pattern  similar  to  Fig.  1,  but  with  strands  of  only  half  the 
v/idth  of  those  there  represented.     The  rectangles  shown  will  be  of  only 
half  the  length  of  those  shown  in  Fig.  1. 

2.  Draw  a  pattern  similar  to  Fig.  2,  but  with  strands  of  only  half  the 
width  of  those  there  represented. 

3.  Draw  a  pattern  similar  to  Fig.  ?>,  but  make  the  lines  of  every  figure 
contained  in  it  twice  the  length  of  those  there  represented.     How,  then, 
will  the  area  of  each  of  the  figures  compare  with  the  area  of  a  similar  fig- 
ure in  the  copy  ? 

4.  Draw  a  pattern  similar  to  and  arranged  like  Fig.  G,  but  make  the  di- 
agonal squares  one  quarter  of  the  area  of  those  there  represented,  and  the 
octagons  only  three  spaces  in  height  and  three  in  width.     Let  the  borders 
of  the  figures  be  only  straight  lines.     Give  to  the  primary  diagonal  squares 
the  dotted  shading,  and  leave  the  octagons  unshaded.     This  will  form  a 
handsome  oil-cloth  pattern. 

5.  Draw  a  pattern  similar  to  Fig.  7,  and  of  the  same  proportions,  but 
containing  an  area  4  times  that  of  the  copy. 

6.  Draw  a  pattern  similar  to  Fig.  8,  and  of  the  same  proportions,  but  con- 
taining an  area  4  times  that  of  the  copy. 

7.  Draw  Fig.  9,  extended  upward,  but  make  dark  diagonal  and  smaller 
squares  in  place  of  the  dark  erect-  squares  now  shown  ;  then  draw  the  same 
pattern  on  the  right,  and  also  on  the  left,  touching  at  the  extreme  angles, 
so  as  thus  to  cover  the  whole  paper  with  a  harmonious  pattern. 

Free-hand  drawing  of  the  figures  of  page  4,  and  of  all  the 
problems  except  the  1th,  on  the  blackboard. 

PAGE  FIVE. 

Figs.  11, 12, 13,  and  14  are  plain  border  patterns;  11  and 
12  being  forms  of  the  Grecian /;•<#,  to  be  noticed  hereafter. 

Figs.  15, 16, 17, 18, 19,  and  20  are  representations  of  flat 
braid  of  3, 4, 5, 7, 9,  and  1 1  strands.  In  Fig.  1 6  the  next  move- 
ment is  to  turn  the  a  strand  upward,  break  it  down  on  the 
dotted  line  1  2,  and  pass  it  over  1)  and  under  c.  Then  break 
the  strand  d  downward  on  the  line  3  4-,  and  pass  it  under  a, 
and  so  on  continuously. 

In  Fig.  17  the  movement  is  continuously  from  the  out- 
side, over  one  and  under  one  /  in  Fig.  18,  over  one  and  under 
two,  beginning  on  the  left ;  in  Fig.  19,  beginning  on  the  left, 
over  one,  under  two,  and  over  one;  in  Fig.  20,  beginning  on 
the  left,  over  one,  under  two,  and  over  two. 


STEAIGHT   LINES   AND   PLANE    SURFACES.  69 

Fig.  21  is  a  pattern  of  interlacing  diagonal  net-work,  em- 
bracing diagonal  squares  that  are  distinguished  by  three 
forms  of  shading  or  coloring. 

Fig.  22  represents  an  embroidered  pattern  brought  a  few 
years  ago  from  the  East  Indies.  Here  \\\Q  forms  alone  can 
be  given,  as  the  colors  can  not  be  represented.  In  the  orig- 
inal pattern  the  four  stars  of  each  cross-shaped  figure  are 
white  or  silver,  on  a  black  ground  inclosed  by  a  silver  line ; 
and  the  small  dark  squares  and  the  straight  lines  connect- 
ing them  are  golden. 

Fig.  23  is  the  filling  up  of  a  mosaic  pattern  of  Byzantine 
pavement.  The  numerous  symmetrical  figures  that  may  be 
discerned  in  it  show  both  the  intricacy  and  at  the  same  time 
the  harmonious  simplicity  of  the  Byzantine  style.  By  the 
aid  of  the  ruled  paper  similar  patterns  of  almost  endless  va- 
riety may  be  designed. 

For  free-hand  drawing  on  the  blackboard  take  Figs.  13, 14, 
15,16,17,  and  18.  They  may  be  shaded  slightly  with  col- 
ored chalks,  so  as  to  make  the  interlacings  plain. 

PAGE  SIX. 

Fig.  24  is  the  simple  generating  form  of  the  Grecian  sin- 
gle fret,  or  meander — a  species  of  architectural  ornament 
consisting  of  one  or  more  small  projecting  fillets,  or  rectan- 
gular bands,  meeting,  originally,  in  vertical  and  horizontal 
directions  only.  Although  this  ornament  was  originated 
by  the  Greeks,  quite  similar  rudimentary  forms  of  the  fret 
have  been  found  among  the  Chinese  and  the  Mexicans.  The 
Arabians  extended  the  Greek  fret  to  diagonal  and  curved 
interlacing  bands;  and  the  Moors  afterward  extended  it  to 
that  infinite  variety  of  interlaced  ornaments,  formed  by  the 
intersection  of  equidistant  diagonal  lines,  which  are  so  con- 
spicuous a  feature  in  the  ornamentation  of  the  Alhambra. 
In  addition  to  the  most  important  of  ihz  plane  surface  Gre- 
cian frets,  here  given,  and  some  of  the  Moorish  that  are  best 
adapted  to  drawing  purposes,  we  have  also  shown  several 
of  them  in  the  second  number  of  the  Drawing  Series,  in  their 
more  natural  form  in  architecture,  as  solids. 

Fig.  24  requires  no  directions  for  drawing  it.     Fig.  11,  on 


70  INDUSTRIAL    DRAWING.  [i^OK    NO.  I. 

page  5,  is  the  same  as  this,  with  the  exception  that  Fig.  11 
has  an  interlacing  band  running  centrally  through  it.  The 
ruler  may  be  used  for  all  the  drawings  on  this  page ;  but 
the  shading  of  the  darker  parts  (by  India  ink)  should  be 
lighter  than  the  copies. 

Fig.  25  is  a  single  fret,  with  the  band  returning  upon  it- 
self at  regular  intervals.  In  drawing  the  frets,  draw  the 
shaded  portions  only,  and,  as  you  proceed,  trace  a  very  faint 
dotted  line  through  the  central  part  of  the  fret,  to  distin- 
guish it  from  the  unshaded  intermediate  spaces.  The  frets 
are  best  shaded,  mainly,  by  India  ink  ;  but  where  there  are 
two  interlacing  bands,  one  of  them  should  have  the  running 
dot  shading. 

Fig.  26  is  also  a  single  fret,  a  little  more  complicated  than 
the  former  two. 

Fig.  27  is  a  double  fret,  formed  of  two  interlacing  bands. 
A  single  band  should  first  be  drawn  throughout,  tracing  it 
lightly  at  first ;  the  spaces  for  the  other  band  will  then  be 
readily  apparent. 

Fig.  28  is  a  double  fret,  formed  by  one  single  fret  backing 
upon  another  single  fret  of  the  same  form. 

Fig.  29  is  an  interlacing  double  fret.  Trace  one  of  the 
bands  throughout  very  lightly  before  beginning  with  the 
other,  so  as  not  to  interfere  with  the  crossings.  The  ruler 
should  not  be  used  (if  at  all)  until  the  entire  fret  is  clearly 
but  lightly  marked  out  with  the  pencil  alone.  Observe 
that,  in  all  interlacing  fret-work,  any  one  band  passes  alter- 
nately first  over  and  then  under  another. 

Fig.  30  is  the  same  as  Fig.  29,  but  with  spaces  left  be- 
tween the  bands  for  paneling.  Observe  the  vertical  bands 
marked  a  b  in  Fig.  29.  These  are  separated  in  Fig.  30  for 
the  panels,  which,  in  Grecian  architecture,  were  ornamented 
with  various  devices. 

Fig.  31  is  an  interlacing  double  fret,  similar  to  Fig.  30,  in- 
verted end  for  end,- with  spaces  for  ornamental  panels.  In 
all  cases  of  double  frets  it  is  best  to  draw  one  of  the  frets 
throughout  before  beginning  the  other. 

The  fret  here  shown,  with  its  panels,  although  strictly 
Grecian,  was  one  of  the  forms  of  Roman  pavement  that  has 


STRAIGHT  LINES    AND   PLANE    SURFACES.  71 

been  found  in  the  ruins  of  Pompeii.  The  two  bands  com- 
posing the  fret,  which  are  here  differently  shaded,  were. of 
white  marble,  formed  of  the  same  number  of  square  pieces 
as  is  designated  by  the  ruling  of  the  paper ;  and  the  inter- 
mediate spaces,  here  left  unshaded,  were  of  black  marble. 

Fig.  32  is  an  interlacing  double  fret  with  panels. 

Fig.  33  is  a  double  fret  with  panels,  but  is  not  interlacing. 
Take  away  the  panels,  and  the  frets  are  doubly  backed  upon 
one  another. 

Fig.  34  is  an  interlacing  double  fret,  formed  of  distinct 
portions  connected  by  a  rectangular  link. 

Fig.  35  is  a  diagonal  and  horizontal  interlacing  double 
fret;  and,  as  its  form  shows,  is  not  Grecian.  It  is  of  Moor- 
ish origin,  and  is  one  of  the  numerous  kinds  of  complicated 
frets,  painted  in  various  colors,  and  on  variously  colored 
grounds,  on  panels  of  the  walls  of  temples. 

Fig.  36  is  an  interlacing  double  fret,  also  of  Moorish  ori- 
gin. 

For  free-hand  blackboard  exercises  take  Figs.  28, 29, 34,  and 
36.  They  may  be  shaded  lightly. 

PAGE  SEVEN. 

Figs.  37  and  38  are  borders  of  fret -work,  formed  after 
Moorish  and  Arabian  patterns. 

Fig.  39  is  an  Arabian  pattern  of  a  mosaic  pavement,  with 
some  of  the  smaller  subdivisions  omitted.  The  peculiar 
star-form  of  ornamentation  here  shown,  which  is  of  Byzan- 
tine origin,  was  also  used  by  the  Arabians. 

Fig.  40  is  a  diagonal  double  fret,  which  has  been  slightly 
varied  from  an  Arabian  pattern  to  fit  it  to  our  purpose.  In 
copying  it,  either  one  of  the  bands  should  first  be  lightly 
traced  throughout. 

Fig.  41  consists  of  two  four-pointed  stars  interlacing,  so  as 
to  show  an  eight-rayed  or  eight-pointed  star.  In  drawing 
it,  first  take  the  centre,  c,  then  the  four  inner  vertical  and 
horizontal  points  marked  3,  then  the  four  inner  diagonal 
points  marked  2.  Also  take  the  eight  ray  points  in  a  sim- 
ilar manner.  Trace  lines  very  faintly  from  the  outer  to  the 
inner  points;  then  trace  an  inner  set  of  lines  equidistant 


72  INDUSTRIAL    DRAWING.  [BOOK    NO.  I. 

from  these;  after  which  mark  firmly  every  alternate  ray 
border  across  the  other  border  lines,  when,  the  intersections 
being  distinct,  the  whole  can  easily  be  finished.  The  rays 
may  be  made  either  longer  or  shorter  than  those  in  the 
drawing,  it  being  considered  that  two  diagonal  spaces  are 
nearly  equal  to  three  vertical  or  horizontal  spaces. 

Fig.  42  is  copied  from  an  Arabian  pattern  of  a  mosaic 
pavement  in  three  colors ;  white  (or  cream-colored),  red,  and 
black.  The  groundwork  may  be  said  to  consist  of  elon- 
gated hexagons  connected  by  interlacing  diagonal  squares ; 
then  there  is  a  central  interlacing  fret  ingeniously  varied  to 
adapt  it  to  the  other  portions,  so  as  to  make  a  perfectly  har- 
monious meander.  In  drawing  it,  first  trace  the  three  parts 
lightly  in  the  order  here  described. 

Fig.  43  is  also  copied  from  an  Arabian  mosaic,  in  white, 
red,  and  black.  It  is  taken  from  a  pavement  in  Cairo.  It 
will  be  seen  that  the  diagonal  lines  here  are  all  two-space 
diagonals;  and  as  the  drawing  conforms  strictly  to  the  orig- 
inal, it  must  be  true  that  the  original  pattern  was  formed 
by  the  aid  of  precisely  such  horizontal  and  vertical  lines  as 
we  have  used  for  guides  on  the  ruled  paper. 

Observe  how  beautifully  the  nine  small  figures,  in  three 
colors,  and  three  diiferent  forms,  fill  out  the  six -pointed 
star-shaped  figure  at  the  intersection  of  the  several  bands. 
The  entire  pattern  is  a  fine  example  illustrating  the  fund- 
amental principles  of  decoration ;  that  all  ornament  should 
be  based  upon  a  geometrical  construction,  and  that  every 
pattern  should  possess  fitness,  proportion,  and  harmony,  the 
result  of  all  which  will  be  a  feeling  of  satisfied  repose,  with 
which  every  such  decoration  will  impress  the  beholder, 
leaving  nothing  further  to  be  desired  within  the  scope  of 
the  ornamentation. 

For  free-hand  blackboard  exercises  take  Figs.  37,  38,  and 
41.  Observe  the  heavy  shading  on  those  sides  that  would 
be  in  shade  if  the  light  came  in  the  direction  indicated  by 
the  arrow. 


CURVED   LINES   AND   PLANE    SURFACES.  73 


III.  CURVED  LINES  AND  PLANE  SURFACES. 

PAGE  EIGHT. 

A  curved  line  is  one  which  is  continually  changing  its  di- 
rection. If  the  curve  be  uniform,  it  forms  part  of  the  cir- 
cumference of  a  circle. 

A  circle  is  a  plane  bounded  by  a  single  curved  line  called 
its  circumference,  every  part  of  which  is  equally  distant 
from  a  point  within  it  called  the  centre.  The  circumference 
itself  is  usually  called  a  circle.  A  straight  line  drawn  from 
the  centre  to  any  part  of  the  circumference  is  called  a  ra- 
dius. 

Fig.  I.  At  A  are  six  uniform  curves  of  five  spaces'  span 
(five  inches),  and  a  depth  of  one  space ;  and  at  Fig.  1  this  curve 
forms  part  of  a  perfect  circle.  At  a  and  b  the  directions 
of  the  curves  are  changed ;  but  all  combined  form  a  harmo- 
nious and  equally  balanced  figure,  because  the  additions  a 
and  b  are  uniform  in  position  and  curvature.  These  figures 
should  be  drawn  with  the  compasses,  using  the  pencil  to 
make  the  connections  of  the  curves  uniform.  Let  the  pupil 
find  the  centres  from  which  the  curves  a  and  b  are  struck. 

Fig.  2  is  formed  of  the  same  pattern  curve  used  in  differ- 
ent positions,  but  all  combined  to  form  a  harmonious  figure. 
If  either  of  the  half  curves,  c  or  d,  were  omitted,  or  changed 
in  position,  the  harmony  of  the  figure  would  be  destroyed. 
At  Yihe  same  form  of  curve  is  used.  Let  the  pupil  find  the 
centres  from  which  the  curves  are  struck. 

Fig.  3  is  also  a  harmonious  figure,  described  wholly  by  the 
compasses ;  but  the  inner  border  lines  from  e  to  h  and  from 
g  to  f  are  described  with  a  less  radius  than  that  used  for 
the  other  curves.  The  curves  e  i  and  g  %  are,  each,  only  half 
of  the  pattern  curve,  and  are  described  from  the  points  1 
and  2. 

Fig.  4.  At  _Z?  is  another  pattern  curve  representing  a  span 
of  six  inches  and  a  depth  of  one  inch,  described  from  the 
centre  c,  with  a  radius  of  five  inches.  In  the  shaded  four- 
angled  figure  the  pattern  curve  is  used  in  four  different  po- 

D 


74  INDUSTRIAL   DRAWING.  [BOOK   NO.  I. 

sitions.  The  centres  from  which  these  curves  are  described 
are  easily  obtained  on  the  ruled  paper. 

Fig.  5  is  formed  wholly  of  combinations  of  the  pattern 
curve  .Z?,  with  two  half  curves  at  the  base,  which,  however, 
are  not  described  from  the  same  centres  as  the  curves  with 
which  they  unite. 

Fig.  6  is  also  formed  of  combinations  of  the  pattern  curve 
JB.  Remembering  that  all  these  curves  are  described  with 
a  radius  of  five  spaces  (or  five  inches),  it  will  be  easy  for  the 
pupil  to  find  their  centres. 

Fig.  7.  We  have  here,  at  (7,  a  new  pattern  curve,  of  a  span 
of  three  spaces,  and  a  radius  of  one  space  and  a  half.  This 
curve  forms  more  than  a  quarter  of  the  circumference  of  a 
circle,  as  may  be  seen  in  the  completed  circle  at  b. 

Fig.  8  is  formed  by  very  simple  combinations  of  the  en- 
tire pattern  curve  C. 

Fig.  9  is  formed  by  adding,  in  Fig.  8,  portions  of  the  pat- 
tern curve  to  the  upper  and  lower  extremities. 

From  the  foregoing  figures  it  will  be  seen  that  we  may 
take  different  portions  of  any  one  regular  curve,  and  com- 
bine them  in  a  great  variety  of  harmonious  patterns.  It  is 
only  to  a  very  limited  extent,  however,  that  we  can  combine 
uniform  curves  of  different  radii  in  the  same  pattern,  with- 
out destroying  that  gracefulness  of  form  which  is  required 
to  please  the  eye,  and  give  to  the  mind  a  feeling  of  repose 
in  the  contemplation. 

We  now  come  to  the  consideration  of  irregular  curves, 
such  as  can  not  be  drawn,  to  any  great  extent,  by  the  aid  of 
compasses. 

Fig.  10.  We  have  here  a  bell-shaped  figure,  drawn  uni- 
formly on  both  sides  of  the  central  and  balancing  line  a  b. 
We  must  draw  one  side  by  the  eye  alone,  and  give  to  the 
waving  line  as  graceful  a  curvature  as  we  can  ;  and  a  great 
many  different  forms  and  proportions  will  answer  the  re- 
quirements of  graceful  curvature ;  but  the  line  must,  never- 
theless, be  such  as  will  please  a  cultivated  eye.  Having, 
therefore,  the  point  x  and  the  central  line  a  b,  we  connect  a 
and  x  by  a  curved  line  that  pleases  the  eye.  If,  now,  we 
can  draw  a  line  exactly  like  it  on  the  other  side  of  a  #,  we 


CURVED   LINES    AND   PLANE    SURFACES.  <"5 

shall  have  a  figure  of  harmonious  form,  whether  a  a;  be  the 
most  graceful  line,  by  itself,  that  could  be  drawn,  or  not. 
But  if  a  x  should  be  drawn  of  the  most  graceful  form  pos- 
sible, and  a  y  just  as  graceful  in  itself,  but  differing  from  u 
x,  the  combination  of  the  two  graceful  forms  would  be  dis- 
cordant, and  make  an  inharmonious  figure,  because  wanting 
in  symmetry  of  parts. 

Having,  therefore,  a  x,  we  designate  in  it  any  number  of 
points  in  which  it  crosses  either  the  horizontal  or  vertical 
lines  of  the  paper.  Let  1,  2,  3,  4,  and  5  be  these  points. 
Then  dot,  lightly,  the  corresponding  points  1, 2, 3, 4, 5  on  the 
other  side  of  a  b,  and  through  them  trace  the  curve,  at  first 
lightly,  and  afterward  fill  it  out  to  correspond  with  the  line 
a  x.  The  figure  is  thus  made  perfectly  symmetrical.  The 
top  of  the  figure  may  be  either  pointed  or  circular ;  and  the 
bell  may  be  longer,  or  narrower,  or  broader,  or  any  one  out 
of  a  great  variety  of  suitable  proportions ;  yet  if  the  two 
sides  are  alike,  the  figure  will  not  be  unpleasant  to  the  eye. 
The  two  inner  dotted  curves  give  different  proportions  for 
the  bell,  while  the  base  remains  the  same. 

Fig.  11  represents  the  harmonious  outlines  of  a  leaf  form. 
Observe  that  the  border  lines  of  the  leaf  pass  through  the 
points  1,  2,  <•?,  4,  5,  on  each  side  of  the  central  line  c  d,  cor- 
responding to  one  other.  Thus  the  points  1  are,  each,  one 
space  from  the  central  line ;  the  points  £,  each  two  spaces ; 
the  points  3,  each  three  spaces,  etc.  la  this  manner  the  two 
sides  of  the  leaf  are  made  perfectly  symmetrical.  The  radi- 
ations of  the  veins  from  the  midrib  of  the  leaf  are  also  sym- 
metrical. 

Fig.  12  is  another  form  of  leaf,  and  although  very  differ- 
ent from  Fig.  11  is  equally  harmonious  in  proportions.  The 
pupU  should  now  have  no  difficulty  in  copying  it  accurately 
on  the  ruled  paper. 

Fig.  13.  Here  is  a  group  of  eight  uniform  leaves,  uniform- 
ly radiating  from  a  common  centre.  The  four  points  a,c, 
€,  (7,  are,  each,  four  spaces  from  the  centre,  and  on  lines  at 
right  angles  with  one  another.  Hence  these  four  points  are 
symmetrically  arranged.  The  other  four  points,  £,  d,  /,  A, 
are  in  the  diagonals,  which  are  intermediate  between  the 


76  INDUSTRIAL   DRAWING.  [BOOK   NO.  I. 

other  lines,  and  are,  each,  at  the  distance  of  three  diagonals 
from  the  centre,  a  trifle  greater  than  the  distance  of  the 
other  four  points,  but  sufficiently  accurate  to  give  the  figure 
all  the  symmetry  required.  Then  each  leaf  must  be  drawn 
by  hand.  At  D  the  method  of  beginning  the  drawing  is 
shown ;  at  E  it  is  carried  still  farther  forward ;  and  in  Fig. 
13  it  is  completed. 

PROBLEMS   FOR   PRACTICE. 

1.  Draw  Fig.  1,  and  draw  within  it  a  curved  line  one  space  distant  from 
the  outer  border  line. 

2.  Draw  Fig.  2  in  the  same  manner. 

3.  Draw  Y  with  an  outer  border  line  one  space  distant  from  the  inner 
border  line,  and  tint  the  space  between  the  two. 

4.  Draw  Fig.  3  inverted,  with  the  upper  part  downward. 

5.  Draw  the  upper  part  of  Fig.  5  on  each  of  the  four  sides  of  the  square 
H,  so  that  it  shall  project  in  four  directions  from  the  central  square. 

6.  Draw  Fig.  10  of  the  same  length  as  at  present,  but  with  a  base,  x  y,  of 
ten  spaces. 

7.  Draw  Fig.  11  eight  spaces  broad  at  3  3. 

8.  Draw  Fig.  12  four  spaces  broader  than  in  the  book,  and  give  to  the  fig- 
ure a  graceful  curve  throughout. 

Blackboard  Exercises. — Figs.  1  to  9  inclusive.  For  these 
a  pair  of  chalk-crayon  compasses  will  be  needed.  See  di- 
rections, page  49. 

PAGE  NINE. 

Fig.  14.  The  circle  here  drawn  from  the  centre  c  has 
longitudinal  divisions,  by  pure  curves  drawn  from  different- 
points  on  the  central  vertical  line. 

Fig.  15.  The  irregular  curve  on  the  left  of  the  central  line 
a  b  had  first  to  be  drawn  by  the  eye — the  design  being  to 
form  a  graceful,  well-proportioned  curve.  Its  counterpart, 
on  the  other  side  of  the  line,  is  to  be  drawn  in  the  manner 
directed  for  drawing  the  preceding  figures,  10, 11,  and  12. 

Fig.  16  is  Fig.  15  completed  in  outline;  the  two  portions 
on  each  side  of  the  central  line  a  b  being  made  symmetrical. 

Fig.  17,  the  completed  form,  is  a  portion  of  a  Pompeian 
border  ornament,  painted  in  yellow  on  a  black  ground.  The 
central  figure  is  proportioned  symmetrically  also.  The 
whole  is,  doubtless,  of  Greek  origin. 


CURVED   LINES   AXD  PLANE   SURFACES.  77 

Figs.  18  and  19  are  symmetrically  formed  leaves,  the 
drawing  of  which  now  requires  no  farther  explanation.  Ob- 
serve how  perfectly  the  one  side  corresponds  to  the  other, 
and  how  easy  it  is  to  draw  such  forms  on  the  ruled  paper. 

Fig.  20,  a  common  form  of  Grecian  conventional  leaf  or- 
namentation, is  arranged  symmetrically  on  each  side  of  a 
vertical  central  line. 

Fig.  21,  a  conventional  leaf  form  of  ornamentation,  can 
now  be  easily  drawn. 

Fig.  22,  also  a  leaf  form,  is  (or  should  be)  equally  bal- 
anced on  each  side  of  the  line  a  b.  Observe  that  the  points 
c  and  d  are  at  equal  distances,  horizontally,  from  the  line  a 
by  so  also  the  points  e  and/*,  and  g  and  h ;  and  each  leaf  di- 
vision on  one  side  must  have  a  corresponding  division  on 
the  other  side. 

Fig.  23,  a  Grecian  pattern,  shows  one  of  the  many  forms 
of  conventional  leafage  found  on  Grecian  vases.  The  leaf 
forms  which  the  Greeks  (and  all  the  ancients)  used  in  their 
ornamentation  were  far  removed  from  any  natural  types; 
and  while  they  are  constructed  on  the  general  principles 
which  reign  in  all  plants,  they  never  attempt  to  represent 
any  particular  species.  The  moderns,  departing  from  the 
true  principles  of  ornamentation  which  prevail  in  ancient 
art,  attempt  to  give  representations  of  real  leaves,  flowers, 
etc.,  being  even  guilty  of  the  anomaly  of  sculpturing  the 
most  delicate  flowers  in  stone ! 

Observe  that  the  two  portions  of  each  pattern  figure  are 
arranged,  symmetrically,  on  each  side  of  a  central  line  (as  a 
o),  which  equally  divides  not  only  the  inclosing  border,  but 
the  symmetrical  group  of  leaves  also.  Therefore  one  half  of 
each  group  is  first  to  be  drawn,  and  then  the  other  half  is  to 
be  made  to  correspond  to  it,  as  shown  in  preceding  figures. 
The  pupils  should  trace  the  leaf  forms  very  lightly  at  first. 

Fig.  24  is  the  form  of  a  Grecian  ornament,  painted  in  gold 
on  a  blue  ground,  on  a  sunken  panel  of  the  ceiling  of  a  Gre- 
cian temple.  Here,  also,  it  will  be  observed,  the  leafage  is 
purely  conventional  in  form,  and  not  designed  to  represent 
any  particular  group  in  nature. 

The  pattern  is  arranged  symmetrically  around  the  centre  c, 


78  INDUSTRIAL   DRAWING.  [BOOK   NO.  I. 

the  diagonal  lea£blades  dividing  it  into  four  equal  portions. 
Taking  the  upper  quarter  section,  observe  that  the  points 
of  the  leaves  b  b  are  equidistant  from  the  central  line ;  the 
slender  volutes  bend  over  equally  in  graceful  curves,  and 
the  radiating  leafage  below  them  fills  the  figure  with  grace- 
ful forms  symmetrically  disposed.  Having  the  ruled  paper, 
the  pattern  can  be  accurately  drawn  without  difficulty. 

PROBLEMS   FOB  PRACTICE. 

After  the  pupil  has  copied  the  examples  on  page  9,  let  him  draw  the  fol- 
lowing : 

1 .  Draw  a  figure  similar  to  Fig.  1 7,  but  two  spaces  higher,  and  two  spaces 
naiTOwer. 

2.  Draw  Figs.  18  and  19,  each  four  spaces  longer,  and  of  the  same  width, 
in  the  widest  parts,  as  in  the  book. 

3.  Draw  Fig.  21,  symmetrically,  on  a  smaller  scale. 

4.  Let  the  pupil  design  a  figure  similar  to  Fig.  22,  but  of  different  propor- 
tions. 

5.  Let  the  pupil  draw  the  pattern  figures  in  Fig.  23,  each  two  spaces  high- 
er, but  of  the  same  width  as  at  present. 

Blackboard  Iticercises.—Figs.  16, 18, 19, 20, 21,  and  22. 

PAGE  TEN. 

All  the  patterns  on  this  page  are  plain  examples  of  what 
is  called  Renaissance  ornament;  by  which  is  understood 
the  style  of  ornamentation  which  sprung  up  in  Italy  upon 
the  revival  of  art  after  the  Dark  Ages.  (See  page  41.).  The 
examples  here  given  are  from  beautiful  colored  specimens 
of  glazed  pottery  of  the  sixteenth  century. 

Fig.  25.  In  copying  this  pattern  the  outlines  should  first 
be  drawn  lightly,  as  shown  at  A.  First  draw  a  single-line 
outline  of  all  the  figures  in  the  pattern,  making  each  figure 
symmetrical  on  each  side  of  its  central  line;  then  finish  the 
two-line  border  of  each  figure.  The  minor  divisions  of  each 
figure  can  then  easily  be  drawn. 

Fig.  26.  The  outlines  of  this  figure  are  to  be  drawn  by 
the  compasses,  as  shown  at  B.  Thus,  the  points  a  are  the 
centres  of  the  semicircles  which  make  up  the  general  outline 
of  the  pattern. 

Fig.  27.  In  this  pattern  the  outer  semicircle  of  each  fig- 
ure is  drawn  from  the  centre  a,  with  a  radius  of  five  spaces ; 


CURVED   LINES   AND   PLANE    SURFACES.  79 

and  the  inner  and  smaller  carves  are  described  from  the 
points  b,  cy  d,  etc. 

Fig.  28.  While  the  lower  portion  of  this  bulb  pattern  is  to 
be  drawn  with  the  compasses,  they  can  afford  no  aid  in 
drawing  the  upper  portion.  The  outlines  of  the  central 
row  of  bulbs  are  first  to  be  drawn  of  uniform  curves  on 
both  sides,  for  which  the  ruling  will  be  a  perfect  guide. 

PROBLEMS   FOR  PRACTICE. 

1 .  Draw  and  shade  Fig.  25  in  full,  after  the  enlarged  outline  form  shown 
atZ>. 

2.  Draw  and  shade  Fig.  26,  according  to  the  enlarged  outline  form  shown 
at  (7,  which  is  four  times  the  size  of  the  pattern.     Why  is  it  four  times  the 
size? 

3.  Draw  an  outline  of  one  of  the  series  in  Fig.  27,  four  times  its  present  size. 

4.  Draw  Fig.  28  lengthwise  of  the  paper,  and  four  times  its  present  size. 

Blackboard  Exercises. — The  bulb-pattern  Fig.,  and  Fig. 
25,  enlarged  as  at  D. 

PAGE  ELEVEN. 

Fig.  29  is  a  Grecian  pattern  of  a  painting  on  vases.  It  is 
still  another  example  of  conventional  flower  representation, 
in  which  only  the  general  principle  of  the  pendulous  flower- 
bud  is  retained.  It  is  drawn  with  the  greatest  regard  to 
symmetry.  The  right-hand  portion,  left  partly  unshaded, 
shows  how  the  forms  of  the  pendulous  buds  are  drawn  sym- 
metrically, and  of  uniform  size. 

Fig.  30  is  a  Byzantine  interlaced  circular  pattern,  sculp- 
tured in  stone,  from  Milan,  Italy.  It  is  very  easily  drawn, 
almost  wholly  by  the  compasses.  Trace  the  whole  very 
lightly  at  first.  The  light  is  supposed  to  come  diagonally 
from  above,  from  the  right  hand. 

Fig.  31  is  a  copy  of  one  of  the  Assyrian  painted  ornaments 
found  among  the  ruins  of  Nineveh.  The  central  circles,  a 
(see  the  end  marked  -S),  were  black ;  the  inclosing  ring,  by 
dark  reddish  brown ;  the  winding  band,  c,  orange ;  and  the 
other  winding  band,  d,  blue ;  while  the  spaces  within  the 
borders,  outside  of  the  winding  bands,  had  a  groundwork 
of  reddish  brown.  The  blue  winding  band  was  separated 
from  the  orange,  wherever  they  were  contiguous,  by  a  black 


80  INDUSTRIAL  DKAWIXG.  [BOOK  NO.  L 

line.  Thus  the  ancient  Assyrians  understood  the  now  ad- 
mitted principle  that  one  color  should  never  impinge  upon 
another,  and  that  all  contiguous  colors  should  be  separated, 
generally  by  either  white  or  black  lines.  The  entire  pattern 
may  be  easily  and  accurately  drawn  by  the  compasses. 
The  same  pattern  is  also  found  differently  lined  and  colored, 
somewhat  as  shown  at  the  end  marked  C. 

Fig.  32  is  a  Byzantine  pattern  of  interlaced  ornament,  to 
be  drawn  wholly  by  the  compasses.  The  central  points 
from  which  the  several  circles  and  semicircles  are  described 
can  easily  be  found  by  the  pupil. 

Fig.  33  is  a  partially  completed  pattern,  showing  the  meth- 
od of  putting  in  a  series  of  uniform  divisions  that  radiate 
from  the  centre  of  a  circle.  The  circle  is  easily  divided  into 
eight  equal  parts  by  lines  radiating  from  the  centre  on  the 
vertical,  horizontal,  and  diagonal  lines,  as  indicated  by  the 
lettering.  Each  eighth  of  the  circumference  is  then  to  be 
divided  into  three  equal  parts  by  the  compasses,  and  the 
opposite  points  in  the  circumference  are  connected  by  slight- 
ly traced  or  dotted  lines  passing  through  the  centre.  The 
bases  of  the  radiating  white  stars  can  then  be  designated 
with  sufficient  accuracy  by  the  eye,  as  the  points  to  which 
the  sides  of  the  stars  are  to  be  drawn,  on  the  second  circle, 
are  intermediate  between  the  radiating  dotted  lines.  Fig- 
ures similar  to  this  are  numerous  in  ancient  ornamental  art. 

Fig.  34  is  a  Byzantine  pattern  of  interlacing  circles. 
Though  seemingly  intricate,  it  is  quite  easily  drawn,  and 
wholly  by  the  compasses.  It  would,  however,  be  impossi- 
ble to  draw  it  with  any  approach  to  accuracy  without  the 
aid  of  the  ruled  paper ;  and  there  can  be  no  doubt  that  it 
was  originally  drawn  on  a  ground  prepared  with  lines  such 
as  those  we  have  given.  The  ruling  gives  the  exact  centre 
of  every  circle,  and  renders  all  measuring  unnecessary.  Ob- 
serve that  the  heavy  shadows  indicate  that  the  light  comes 
from  above,  and  from  the  left.  The  background  may  be 
shaded  with  a  uniform  tint  of  India  ink,  or  by  the  pencil, 
as  indicated  in  the  upper  left-hand  portion. 

Fig.  35  is  also  a  Byzantine  interlaced  ornament,  a  consid- 
erable portion  of  which  may  be  drawn  by  the  compasses. 


CURVED   LINES   AND   PLANE    SUEFACES.  81 

Thus  the  three  circular  bands,  and  the  projections  of  the  in- 
terlacing loops  as  described  from,  the  vertical,  horizontal, 
and  diagonal  centres  a,  a,  a,  etc.,  are  all  easily  and  accurate- 
ly drawn.  The  connecting  and  interlacing  of  the  loops  cen- 
trally must  be  done  by  hand,  guided  by  the  eye  alone ;  but 
after  having  drawn  one  of  the  connections,  as  from  b  to  d, 
the  others  may  be  drawn  in  symmetrical  conformity  to  it. 
Observe  how  the  shadows  are  cast,  the  light  coming  diag- 
onally from  the  left,  and  from  above. 

PROBLEMS   FOR   PRACTICE. 

1.  Draw  a  pattern  similar  to  Fig.  29,  but  let  the  semicircular  black  line, 
from  the  end  of  which  the  flower-bud  is  suspended,  be  drawn  with  a  radius 
of  four  spaces,  instead  of  three ;  and  let  the  flower-bud  be  two  spaces  longer 
and  one  space  broader  at  the  broadest  part. 

2.  Draw  a  pattern  similar  to  Fig.  31,  but  describe  the  larger  circles  with 
a  radius  one  space  greater  than  in  the  figure. 

3.  Draw  a  pattern  similar  to  Fig.  33,  but  describe  the  outer  circle  with 
a  radius  of  eight  spaces,  and  divide  each  eighth  part  of  the  circumference 
into  four  equal  parts,  instead  of  three,  thus  giving  to  each  eighth  part  one 
ray  more  than  in  the  figure. 

4.  Draw  Fig.  34  on  a  larger  scale,  having  the  radius  of  the  larger  circles 
six  spaces,  instead  of  five. 

5.  Draw  Fig.  35  on  a  larger  scale,  at  the  option  of  the  pupil. 

Blackboard  Exercise.  —  So  much  of  Fig.  35  as  there  is 
room  for  on  the  board.  It  would  require  a  board  four  feet 
square  to  draw  it  in  accordance  with  the  enlarged  ruling 
of  the  board.  .#£ 

PAGE  TWELVE. 

We  have  given  on  this  page  a  number  of  original  designs, 
for  the  purpose  of  indicating  the  facility  with  which  an  im- 
mense variety  of  very  pleasing  patterns  may  be  drawn  al- 
most wholly  by  the  compasses  alone. 

Fig.  36  is  an  ornamental  figure  called  a  quarterfoil,  in- 
closed within  a  circle.  The  quarterfoil,  often  used  in  archi- 
tecture, is  disposed  in  four  segments  of  circles,  and  is  a  con- 
ventional representation  of  an  expanded  flower  of  four  pe- 
tals. The  lettering  shows  with  what  ease  and  accuracy  it 
is  drawn  on  the  ruled  paper. 

Fig.  37  is  an  ornamental  figure  consisting  of  eight  e°' 
D2 


82  INDUSTRIAL   DRAWING.  [BOOK   NO.  I. 

ments  of  circles,  described  from  the  eight  numbered  points 
which  are  at  the  extremities  of  the  dotted  vertical,  horizon- 
tal^ and  diagonal  lines.  After  drawing  these  dotted  lines^, 
of  indefinite  extent,  but  all  passing  through  the  centre  c, 
equidistant  points  in  all  of  them,  for  describing  the  seg- 
ments, may  be  found  by  cutting  the  lines  with  the  circum- 
ference of  a  circle  described  from  the  centre  c.  From  the 
points  ./,  #,  3,  4>  etc.,  the  segments  are  described  with  a 
length  of  radius  that  will  barely  allow  the  inner  segments 
of  circles  to  touch  one  another. 

Fig.  38.  The  segments  of  circles  are  here  drawn  in  a 
manner  similar  to  those  shown  in  Fig.  37;  and  then  these 
segments  are  connected  by  ribs  passing  through  the  centre. 
The  intervening  spaces  are  then  shaded,  so  as  to  give  the 
filling -up  a  raised  and  rounded  appearance,  the  heaviest 
shades  being  found  on  the  lower  right-hand  portions — the 
light  being  indicated  by  the  arrow  A.  as  coming  diagonally 
from  the  left  hand,  and  from  above.  The  spaces  covered 
by  the  heavy  dark  shade  on  the  outside  borders  of  the  fig- 
ure— the  heaviest  at  the  right  and  below — are  marked  out 
accurately  by  the  compasses,  by  moving  the  fi^ed  point  of 
the  compasses  uniform  distances,  diagonally,  from  the  cen- 
tres 1  to  the  points  2.  By  looking  at  the  figure  through  a 
tubular  opening,  it  will  seem  to  stand  out  from  the  paper  as 
if  embossed  upon  it — in  relief. 

Fig.  39  is  similar  in  its  border  outlines  to  Fig.  38;  and 
all  of  it,  with  the  exception  of  the  central  portion,  may  be 
easily  and  accurately  drawn,  and  mostly  shaded,  by  the 
compasses  alone.  To  draw  the  central  figure  accurately, 
first  trace  out  lightly,  in  the  space  which  it  covers,  the  ver- 
tical, horizontal,  and  diagonal  lines,  after  which  the  raised 
wedge-shaped  oval  figures  may  be  drawn  in  between 
these  lines  by  hand.  Observe  how  the  heavy  shadows  are 
formed^ 

Fig.  40,  drawn  on  the  same  general  plan  as  the  preceding 

two  figures,  requires  no  additional  explanation.   The  shaded 

cross-ribs  are  easily  drawn  after  making  the  raised  circles 

of  uniform  size  around  their  given  central  points.     As  these 

t~"1  r>oints  are  determined  by  the  tint  lines  on  the  draw- 


CURVED   LINES   AND   PLANE    SURFACES.  83 

ing-paper,  the  whole  figure  may  be  drawn  with  the  greatest 
ease  and  accuracy. 

Fig.  41  is  a  pattern  of  seeming  intricacy,  but  very  easily 
drawn,  and  planned  on  the  same  general  principles  as  the 
preceding  figures  on  the  same  page.  Every  thing  is  drawn 
by  the  compasses  from  points  indicated  by  the  ruling  of  the 
paper,  except  the  raised  wedge-shaped  ovals,  that  look  as  if 
embossed. 

Fig.  42  is  an  interlaced  pattern,  drawn  wholly  by  the  com- 
passes, with  the  exception  of  the  central  figure.  The  figures 
./,  £,  #, 4,  etc.,  show  the  eight  central  points  from  which  the 
interlacing  curves  are  described. 

blackboard  Exwcises. — Figures  41  and  42,  omitting  the 
central  eight-leaf  flower  patterns. 

Page  12  of  this  book  illustrates,  very  happily,  the  great 
advantage  which  the  ruled  paper  affords  for  drawing  curvi- 
linear patterns.  It  would  be  next  to  an  impossibility  to 
draw  these  designs  with  accuracy  without  this  aid;  but 
with  this  ruling,  this  kind  of  drawing,  which  is  used  to  a 
great  extent  in  all  the  decorative  arts,  becomes  a  very  sim- 
ple matter,  easy  of  attainment  by  all  who  can  describe  a 
circle  by  the  aid  of  the  compasses. 


DRAWING-BOOK  No.  II. 


CABINET  PERSPECTIVE— PLANE  SOLIDS. 

THE  Cabinet  Perspective  presented  in  this  series  is  a  meth- 
od of  representing  solids,  both  plane  and  curvilinear,  in 
such  a  manner  that  the  drawings  shall  give,  by  the  aid  of 
the  ruled  paper,  the  correct  measurements  of  the  objects 
represented.  The  ruling  on  the  paper  is  adapted  to  any 
scale  of  measurement ;  but,  for  the  purposes  of  the  present 
illustration,  let  it  be  supposed  that  the  vertical  and  horizon- 
tal lines  on  the  paper  are  respectively  one  inch  apart. 

In  all  drawings  in  what  is  called  Diagonal  Cabinet  Per- 
spective? objects  are  supposed  to  be  viewed  in  a  manner 
similar  to  that  in  which  the  two  cubesf  in  Fig.  1,  on  page  1 
of  Drawing-Book  No.  II.,  are  represented.  Taking  the  cube 
at  JB  for  illustration,  this  may  be  supposed  to  be  a  cube  six 
inches  square,  the  front  face  of  which  is  in  a  vertical  posi- 
tion. The  spectator  is  supposed  to  view  this  cube  from 
such  a  point,  above  and  at  the  right  of  the  cube,  that  he 
may  see  just  as  much  of  the  upper  side  .of  the  cube  as  of  the 
right-hand  side;  so  that  the  apparent  width,  10  9  of  the 
upper  face,  or  12  0  of  the  side  face,  shall  measure,  in  the 
directions  indicated,  one  half  the  width  of  the  front  face ; 
and  so  that  the  diagonal  corner  lines,  1  2, 3  4->  and  5  6,  will 
seem  to  rise  diagonally  at  an  angle  of  forty-five  degrees ; 
while  the  distance  from  which  the  view  is  taken  is  supposed 

*  There  is  a  beautiful  practical  application  of  Cabinet  Perspective,  called 
/Semi-diagonal  Cabinet  Perspective,  which  will  be  found  illustrated  on  pages 
9-11  of  Drawing-Book  No.  IV.  See  page  171. 

t  A  cube  is  a  regular  solid  body,  having  six  equal  square  sides. 


CABINET   PERSPECTIVE PLANE    SOLIDS.  85 

to  be  so  great  that  these  lines  will  appear  to  be,  as  they  are 
here  drawn,  parallel. 

The  front  face  of  the  cube  is  drawn  in  its  real  proportions 
as  a  square,  and  as  though  it  were  seen  in  a  vertical  plane 
directly  fronting  the  spectator.  According  to  the  scale 
supposed  to  be  adopted  in  the  ruling,  the  front  of  the  cube 
measures  six  inches  to  a  side.  The  farther  face  of  the  cube, 
being  also  vertical  and  parallel  with  the  front  face,  and 
therefore  in  a  plane  also  directly  fronting  the  spectator, 
would  also  be  drawn  as  a  square  if  it  could  all  be  seen,  meas- 
uring six  inches  to  a  side.  Hence  each  of  the  lines  2  4  and 
4  6  measure  six  inches,  the  same  as  1 3  and  3  5. 

But  each  of  the  diagonal  lines  1  2,  3  h  and  5  6,  being 
corner  lines  of  the  cube,  must  also  represent  a  measure  of 
six  inches ;  and  as  each  of  these  lines  extends  over  three  di- 
agonals of  the  small  squares,  it  follows  that  what  we  call 
one  diagonal  space  measures  twice  as  much  as  a  vertical  or 
a  horizontal  space,  whenever  this  diagonal  space  is  applied  to 
the  measurement  of  a  line  representing  a  horizontal  line. 
We  may,  therefore,  adopt  the  following  rule  for  the  meas- 
urement (and  also  for  the  drawing)  of  all  objects  in  diag- 
onal cabinet  perspective. 

ELEMENTARY  RULE. 

Drawings  of  surf  aces  that  are  supposed  to  be  in  a  vertical 
plane  fronting  the  spectator  are  measurable,  in  any  direc- 
tion, according  to  the  scale  adopted  for  the  vertical  and  hor- 
izontal spaces  of  the  ruling;  while  each  DIAGONAL  space 
fhat  measures  a  line  in  a  horizontal  and  diagonal  position  is 
to  be  taken  as  TWICE  the  measure  of  a  space  of  the  other  kind. 

Applying  this  rule  to  the  cube  at  2?,  Fig.  1,  we  find  that 
all  the  horizontal  lines,  and  also  all  the  vertical  lines  that 
cross  the  front  face  of  the  cube,  measure  each  six  inches  in 
length,  because  each  extends  over  just  six  of  the  ruled 
spaces,  and  all  are  in  a  vertical  plane  fronting  the  spectator. 
For  a  similar  reason  each  one  of  the  vertical  lines  on  the 
right-hand  side  of  the  cube,  and  each  one  of  the  horizontal 
lines  on  the  top  of  the  cube,  measures  six  inches,  because 
each  may  be  supposed  to  be  in  a  plane  like  that  which 


86  INDUSTRIAL   DRAWING.  [BOOK    NO.  II. 

forms  the  front  face  of  the  cube — directly  fronting  the  spec- 
tator. But  such  lines  as  1  2,  3  4,  5  6,  8  9, 10  11,  and  12  IS, 
being  seen  obliquely,  can  not  be  in  any  plane  fronting  the 
spectator;  and  as  they  lie  in  a  diagonal  direction,  and  rep- 
resent horizontal  lines,  they  are  measurable  by  the  prin- 
ciple adopted  in  the  latter  part  of  the  rule.  Hence  the  line 
a  b  measures  five  inches ;  c  d  four  inches ;  g  13  four  inches ; 
m  n  six  inches  :  but  8  9  measures  six  inches ;  10  11  nieas- 
sures  four  inches ;  12  IS  measures  four  inches,  etc.  The 
cube  at  A  measures  two  inches  on  each  of  its  corners. 

Fig.  2.  Applying  the  scale  of  measurement  which  we  have 
adopted  to  the  representation  of  the  cube  at  D,  Fig.  2,  we 
find  that  the  front  face  of  the  cube  is  a  square  of  four  inches 
to  a  side ;  and  that  the  diagonal  horizontal  distance  1  2,  or 
8  4,  or  5  6,  also  measures  four  inches.  Also,  if  we  draw  in- 
termediate lines  between  the  ruled  lines  on  the  paper,  on 
the  upper  face  and  right-hand  face  of  the  cube,  so  as  to  give 
owe-inch  diagonal  measures,  then  sixteen  one-inch  squares 
may  be  counted  on  each  of  the  three  visible  faces  or  sides 
of  the  cube.  We  thus  have,  according  to  the  scale  of  one 
inch  to  a  space  horizontally  or  vertically  between  the  lines, 
and  two  inches  for  a  diagonal  space,  the  exact  measurement 
of  the  three  visible  sides  of  the  cube. 

To  find  the  contents  of  a  cube: 

RULE. — Multiply  the  length  of  a  side  of  the  cube  by  itself, 
and  that  product  again  by  a  side,  and  this  last  product  will 
give  the  contents  required.  (See  Rule  I.,  page  53,  for  the 
measurement  of  surfaces.) 

Thus,  in  the  cube  at  D,  Fig.  2,  if  we  multiply  the  length 
(1  2)  of  one  side,  which  is  four  inches,  by  the  length  (2  4) 
of  another  side,  which  is  also  four  inches,  we  get  the  prod- 
uct 16,  which  is  the  number  of  solid  cubic  inches  contain- 
ed in  the  upper  tier  of  the  cube — as  may  also  be  seen  by 
counting  them ;  and  as  there  are  four  of  these  tiers,  we  mul- 
tiply the  16  by  4,  and  get  64,  the  number  of  cubic  inches  in 
the  four  tiers,  or  in  the  whole  cube.  Or,  4  x  4  x  4  =  64  cubic 
inches. 

One  cubic  inch  is  represented  at  C,  which,  according  to 


CABINET   PERSPECTIVE PLANE    SOLIDS.  87 

the  scale  we  have  adopted  for  page  1,  measures  one  inch  on 
each  of  its  sides ;  and  at  E  are  sixteen  cubic  inches,  equal 
to  the  upper  tier  in  D. 

In  straight-line  drawings  in  cabinet  perspective,  the  ruler 
may  be  used  wherever  its  aid  will  give  additional  accuracy 
to  the  drawing. 

The  contents  of  the  cube  .Z?,  Fig.  1,  which  measures  six 
inches  to  a  side,  are  found  by  the  rule  to  be  as  follows. 

Ans.  6x6x6=216  cubic  inches. 

The  drawing  at  E,  Fig.  2,  is  an  example  of  a  parallelopi- 
ped /  a  figure  which  is  defined  as  being  a  solid  whose  faces 
are  six  rectangles,*  the  opposite  faces  being  parallel,  and 
equal  to  each  other.  The  drawing  at  F  also  represents  a 
parallelepiped.  All  squares  are  parallelepipeds;  but  all 
parallelepipeds  are  not  squares. 

To  find  the  contents  of  a  rectangular  (right-angled)  parallelepiped  : 

RULE. — Multiply  the  length  by  the  breadth,  and  that  prod- 
uct by  the  depth,  and  this  last  product  will  give  the  contents 
required. 

The  height  of  the  rectangular  solid  at  E,  Fig.  2,  is  one 
inch  ;  the  breadth  in  one  direction  (1  2)  is  four  inches ;  and 
the  breadth  in  the  other  direction  (3  4)  is  also  four  inches. 
What  are  the  contents?  Ans.  1x4x4  =  16  cubic  inches, 
as  may  be  verified  by  counting  the  small  one-inch  cubes 
which  it  contains. 

The  length  (or  height)  of  the  solid  at  F  is  three  inches  ; 
the  breadth  or  width  is  five  inches  ;  and  the  diagonal  depth 
is  two  inches.  What  are  the  contents  ? 

Ans.  3  x5  x2  =  30  cubic  inches. 

The  two  rules  just  given  may  be  combined  in  one,  as  fol- 
lows: 

To  find  the  contents  of  any  solid  rectangular  figure : 

RULE. — Multiply  the  three  dimensions  together,  and  their 
product  will  be  the  contents  required. 

Fig.  3.  At  6r,  H,  I,  and  e7are  represented  four  parallelo- 

*  In  a  strict  definition  the  sides  need  not,  necessarily,  be  rectangular 
(right-angled)  ;  but  it  is  better,  at  present,  for  the  pupil  in  drawing  to  con- 
sider all  parallelepipeds  as  of  the  rectangular  kind. 


88  INDUSTRIAL   DRAWING.  [BOOK    NO.  II. 

pipeds,  all  of  the  same  size ;  Gr  being  viewed  in  a  vertical 
position,  H horizontally,  and  ^Tand  «7  horizontally  and  diag- 
onally. They  may  be  considered  pieces  of  timber,  each  two 
inches  square  at  the  ends,  and  twelve  inches  long.  What 
are  the  solid  contents  of  each?  Ans.  48  cubic  inches. 

Observe  that,  according  to  the  scale  of  measurement  al- 
ready explained,  these  pieces  measure  precisely  the  same  in 
these  three  different  positions. 

Fig.  4.  According  to  the  definition  of  a  parallelepiped, 
this  figure,  also,  is  one  of  that  kind.  What  are  its  meas- 
urements, and  its  contents  ? 

Observe  that  the  right-hand  sides  of  the  foregoing  figures 
are  represented  as  shaded  with  a  deep  tint  of  India  ink,  the 
front  with  a  lighter  tint,  and  the  top  of  Fig.  4  with  the  run- 
ning dotted  shading.  The  kinds  of  shading  used  in  Fig.  4 
are  well  adapted  to  all  plane  solids,  as  the  object  of  shad- 
ing, in  cabinet  perspective,  is  to  render  the  several  surfaces 
as  marked  and  distinct  as  possible,  one  from  another. 

Fig.  5  is  a  square  frame  composed  of  four  pieces,  each  two 
inches  square;  the  two  diagonal  side-pieces  each  twelve 
inches  long,  and  the  other  two  each  eight  inches  long. 
What  is  the  size  of  the  square  which  they  inclose? 

Fig.  6.  Let  the  pupil  describe  Fig.  6 — that  is,  tell  how 
many  pieces  compose  the  figure,  their  size,  position,  etc. 

Fig.  7  is  drawn,  first,  in  the  same  manner  as  Gr  of  Fig.  3 ; 
it  is  then  divided  so  as  to  represent  cubical  blocks,  each  two 
inches  square,  placed  one  above  another.  Three  of  these 
blocks  are  represented  as  shaded  with  the  hatching  de- 
Hcribed  in  Lesson  IX.  of  Drawing  -  Book  No.  I.,  after  first 
tinting  the  surface  with  India  ink. 

Fig.  8  is  composed  of  two  vertical  blocks,  each  two  inches 
square  and  five  inches  long,  resting  upon  the  ends  of  a  piece 
one  inch  by  two  inches,  and  twelve  inches  long;  the  latter 
being  viewed  diagonally. 

Fig.  9  is  the  same  as  No.  8  inverted,  and  still  viewed  di- 
agonally. 

Fig.  10  is  also  the  same  as  Fig.  8 ;  but  it  is  here  viewed  hori- 
zontally. Thus  the  same  figure  may  be  represented  in  several 
different  positions,  so  as  to  bring  each  side  into  full  view. 


CABINET   PERSPECTIVE — PLANE    £OLIDS.  89 

In  these  several  figures  observe  the  effect  of  the  shading, 
which  is  rapidly  executed  with  different  tints  of  India  ink, 
except  the  upper  surfaces,  which  have  the  running  dot-line 
shading. 

PROBLEMS   FOR   PRACTICE. 

1.  Draw  the  representation  of  a  square  frame  similar  to  Fig.  />,  of  the 
same  outside  and  inside  measure  as  Fig.  5,  but  composed  of  pieces  only  one 
inch  thick  instead  of  two  inches ;  and  all  of  them  two  inches  wide. 

2.  Draw  a  figure  similar  to  Fig.  6,  but  let  the  upper  pieces  be  only  one 
inch  thick  (or  high),  and  let  them  project  each  one  inch  at  both  ends  be- 
yond the  lower  pieces. 

3.  Draw  a  figure  similar  to  Fig.  7  in  position,  but  four  inches  wide  and 
two  inches  thick — the  wide  side  being  in  front. 

4.  Draw  a  figure  similar  in  position  to  Fig.  8,  but  composed  of  three  pieces, 
each  twelve  inches  long,  four  inches  wide,  and  one  inch  thick. 

5.  Draw  Fig.  9  with  the  side  in  a  plane  fronting  the  spectator. 

Free-hand  Blackboard  Exercises. — Figs.  5  to  10  inclusive — 
letting  a  space  on  the  blackboard  represent  its  true  meas- 
ure, two  inches.  In  this  case  only  one  half  of  a  diagonal 
measure  will  be  required  for  a  distance  of  two  inches.  In 
order  to  render  the  several  surfaces  distinct  one  from  an- 
other, the  dark  shades  may  be  represented  by  heavy  ver- 
tical chalk  lines,  and  the  lighter  shades  by  very  light  ver- 
tical lines.  Or,  where  colored  chalk  crayons  are  accessible, 
the  dark  shades  may  be  represented  by  blue  lines. 

Fig.  11,  in  its  complete  outline,  is  a  cube  eight  inches 
square,  having  a  small  cube  two  inches  square  cut  from 
each  of  its  four  upper  corners.  See  the  form  of  the  entire 
cube,  as  represented  by  the  dotted  lines.  What  were  the 
contents  of  the  entire  cube  ?  What  the  contents  after  the 
four  corners  were  taken  out. 

Fig.  12  is  a  cube  twelve  inches  square,  having  a  piece  eight 
inches  square  and  two  inches  thick  taken  from  the  centre  of 
each  of  its  three  visible  sides.  What  were  the  contents  of  the 
entire  cube  before  the  three  pieces  were  taken  out  ?  What 
were  the  contents  after  these  pieces  were  taken  out  ?  Let 
the  pupil  be  careful,  in  drawing  the  figure,  that  his  measures 
shall  be  correct.  Thus,  the  depth  of  the  recess  in  each  side 
must  measure  two  inches.  Thus  c  d  and  m  n  are  measures 


90  INDUSTRIAL  DRAWING.  [BOOK   NO.  II. 

of  two  inches  each ;  and  a  b,  being  one  diagonal,  is  also  a 
measure  of  two  inches. 

Fig.  13  represents  a  rectangular  frame  measuring  ten  by 
fourteen  inches,  and  composed  of  pieces  two  inches  square 
framed  into  posts  two  inches  square  and  ten  inches  long. 

CrifP'  To  get  the  full  effect  of  the  figures  on  this  page, 
view  them  from  a  point  at  the  right,  and  above  them, 
through  the  opening  formed  by  the  partially  closed  hand. 
The  pupil  should  be  accustomed  to  view  his  drawings  in 
the  same  manner. 

PROBLEMS   FOR   PRACTICE. 

1.  Draw  a  figure  similar  to  Fig.  11,  but  twelve  inches  square,  having  a 
block  four  inches  square  cut  from  each  of  its  four  upper  corners. 

What  would  be  the  contents  of  the  entire  cube?  What  the  contents  after 
the  four  comers  were  taken  out? 

2.  Draw  a  figure  similar  to  Fig.  12,  but  fourteen  inches  square,  and  show- 
ing a  rectangular  piece  ten  inches  square  and  four  inches  in  thickness  cut 
from  the  centre  of  each  of  its  three  visible  sides. 

What  would  be  the  contents  of  the  entire  cube?  What  the  contents  after 
the  three  rectangular  pieces  were  taken  out. 

3.  Draw  a  figure  similar  in  all  respects  to  Fig.  13,  except  that  the  four 
horizontal  pieces  of  the  frame-work  are  to  be  only  one  inch  in  veriicr.l 
thickness. 

Free-hand  Blackboard  Exercises. — Figs.  11,  12,  and  13; 
also  the  accompanying  problems. 

PAGE  TWO.— SCALE  OF  ONE  INCH  TO  A  SPACE. 

Fig.  14  represents  a  cube  eight  inches  square;  and  Fig. 
15  represents  a  box  formed  of  one-inch  stuff,  open  at  the  top, 
and  of  a  size  that  will  just  receive  the  cube;  so  that  the  lat- 
ter, when  placed  within  the  box,  shall  fill  it  even  with  the 
top.  At  A  is  the  cover  that  will  just  fit  the  top  of  the  box. 

What  is  the  size  of  the  cover  ?  Of  the  side  J3  of  the  box  ? 
Of  the  sides  C  Cf  Of  the  bottom  of  the  box  ?  What  arc 
the  outside  measures  of  the  box  when  the  cover  is  on  it  ? 

Fig.  16  represents  a  series  of  five  blocks  placed  one  upon 
another,  and  rising  in  the  form  of  stairs.  The  upright  piece 
in  a  stair  (as  a)  is  called  the  riser,  and  the  part  on  which 
the  foot  is  placed  (as  b)  is  called  the  tread  In  Fig.  16  the 
stairs  are  so  placed  that  the  risers  front  the  spectator;  but 


CABINET   PERSPECTIVE — PLANE    SOLIDS.  91 

in  Fig.  17  the  side  of  the  stairway  fronts  the  spectator,  and 
the  riser  is  viewed  diagonally.  Fig.  17  measures  the  same 
as  Fig.  16,  with  the  exception  that  in  Fig.  17  the  lower  block 
is  omitted. 

If  these  two  figures  are  supposed  to  be  drawn  to  a  scale 
of  four  inches  to  a  space,  what  will  be  the  height  of  each 
riser,  and  the  width  of  each  tread  ?  What  the  width  of  the 
stairway  ? 

Fig.  18  represents  a  cabinet  frame-work  formed  of  pieces 
two  inches  square  at  the  ends ;  the  whole  frame  measuring 
ten*  inches  by  twenty-six.  Here  the  longest  side  fronts  the 
spectator;  but  in  Fig.  19  the  same  frame-work  is  represent- 
ed with  the  end  fronting  the  spectator.  Observe  that  the 
end  below  c  d  of  Fig.  18,  which  is  there  seen  diagonally,  is 
not  seen  at  all  in  Fig.  19  ;  and  that  the  end  below  a  b  of  Fig, 
19  is  not  seen  at  all  in  Fig.  18.  Let  the  pupil  describe  tho 
several  pieces  of  which  the  frame-work  is  composed. 

Fig.  20  represents  a  cross  made  of  pieces  measuring,  at 
the  ends,  two  by  three  inches. 

Fig.  21  represents  a  cabinet  square  made  of  one-by-two- 
inch  stuff,  and  placed  horizontally ;  but,  as  the  spectator  is 
supposed  to  be  above  and  to  the  right  of  it,  one  of  the  arms 
seems  to  rise  at  an  angle  of  forty-five  degrees — being  in  the 
diagonal  direction  of  the  small  ruled  squares. 

Fig.  22  represents  the  same  cabinet  square  in  a  vertical 
position,  and  so  placed  that  the  wide  side  shall  front  the 
spectator.  The  former,  as  placed,  is  a  measure  for  horizon- 
tal and  diagonal  distances ;  the  latter,  for  horizontal  and 
vertical  distances. 

Fig.  23  is  an  upright  frame-work  resting  on  four  blocks, 
each  two  inches  square ;  having  only  two  of  its  sides  and 
the  top  and  bottom  inclosed,  and  containing  three  shelves 
in  addition  to  the  top  and  bottom.  Let  the  pupil  describe 
this  frame-work  more  fully;  and  tell  (or  write  out)  its  meas- 
ures in  all  it's  parts — distances  apart  of  the  shelves,  etc.  The 
whole  on  the  supposition  that  it  is  drawn  to  the  scale  of 
one  inch  to  the  space. 

The  shading  on  this  page  is  supposed  to  be  done,  first, 
with  India  ink;  some  of  the  sides  are  then  shaded  with  lines 


92  INDUSTRIAL   DRAWING.  [BOOK   NO.  II. 

by  the  pencil,  which,  when  not  too  heavy,  give  additional 
life  and  spirit  to  the  drawing. 

PROBLEMS   FOR   PRACTICE. 

As  Fig.  14  represents  a  cube  eight  inches  square,  what  are  its  contents  in 
cubic  inches ?  A ns.  8x8x8=512  cubic  inches. 

As  it  takes  231  cubic  inches  to  make  a  gallon,  how  many  gallons  of  water 
would  the  box,  Fig.  15,  contain  ?  Ans.  2  gallons  and  50  cubic  inches. 

1.  Draw  a  box,  similar  to  Fig.  15,  that  would  contain  a  cube  twelve  inches 
square ;  and  draw  the  cover  separately.     How  many  gallons  of  water  would 
such  a  box  contain  ? 

2.  Draw  a  rectangular  box  open  at  the  top,  the  bottom  of  which  shall 
measure,  on  the  inside,  ten  by  twelve  inches,  and  let  the  box  be  ten  inches 
deep.     How  many  gallons  of  water  will  it  contain  ? 

3.  Draw  a  flight  of  stairs,  similar  to  Fig.  1 7,  to  a  scale  of  six  inches  to 
a  space,  and  having  the  risers  twelve  inches  high,  and  the  tread  eighteen 
inches  wide. 

4.  Draw  a  frame  similar  to  Fig.  18,  but  formed  of  stuff  one  by  two  inches 
— the  one-inch  being  the  height. 

Free-hand  Blackboard  Exercises. — Figs.  17,  18,  and  20, 
and  problems  2  and  3. 

PAGE  THREE.— SCALE  OF  THREE  INCHES  TO  A  SPACE. 

Fig.  24.  At  A  is  represented  a  piece  of  timber  six  by 
twelve  inches  at  the  end,  and  four  feet  long,  broadest  side 
down,  having  a  piece  cut  from  the  centre  of  the  upper  side 
one  foot  square  and  three  inches  deep,  to  receive  crosswise 
nnother  timber  cut  in  like  manner.  At  B  the  two  pieces 
of  timber  are  united  at  right  angles  to  one  another. 

The  remaining  figures  on  this  page  are  examples  of  what 
carpenters  call  scarfing,  which  means  the  uniting  of  two 
pieces  of  timber  longitudinally  by  a  scarf-joint — in  common 
language  often  called  splicing ;  but  the  latter  term  is  more 
properly  applied  to  an  overlapping  joint  that  is  not  notched. 

Fig.  25.  At  C  and  D  are  two  pieces  of  timber  prepared 
for  being  joined  lengthwise  by  a  square  scarf-joint.  After 
placing  the  timbers  in  the  right  position,  they  are  pinned  or 
bolted  together.  Let  the  pupil  describe  the  two  pieces. 

Fig.  26.  At  E  is  the  upper  piece  of  timber,  and  at  F  the 
lower,  to  be  united  by  a  scarf-joint,  which  is  formed  of  the 
square  scarf-joint  combined  with  the  splice-joint.  Observe 


CABINET  PERSPECTIVE — PLANE    SOLIDS.  93 

that  the  line  of  union,  2  4,  in  both  E  and  F,  is  in  the  direc- 
tion of  two-space  diagonals ;  and  as  the  two  lines  are  of 
the  same  length,  and  the  two  timbers  lie  in  the  same  rela- 
tive positions,  the  cut  sections  must  correspond  with  one 
another.  Moreover,  E  and  F  are  redrawn  at  G  and  H; 
and  there  it  is  seen  that  the  diagonal  line  of  union,  2  J^ 
is  in  both  cases  the  diagonal  of  the  same  rectangle  1  2, 
3k 

Fig.  27  shows  two  pieces  of  timber,  Jand  J",  prepared  for 
being  joined  by  the  ordinary  splice-joint.  The  two  tim- 
bers are  to  be  firmly  bolted  together.  But  this  is  by  no 
means  so  firm  a  mode  of  union  as  is  shown  in  Figs.  25 
and  26. 

Fig.  28  shows  a  still  firmer  mode  of  the  scarf-joint  than 
Figs.  25  and  26.  Observe  that  the  lines  of  union  in  K meas- 
ure precisely  the  same,  and  are  in  precisely  the  same  posi- 
tions, as  in  L.  Thus  the  lines  1  2  and  3  4.  are  the  same  in 
length  and  position  as  the  lines  5  6  and  7  8,  both  being 
two-space  diagonals. 

Fig.  29  shows,  perhaps,  the  firmest  of  all  modes  of  the 
scarf-joint,  especially  for  heavy  timbers.  JV  is  the  upper 
piece;  and  the  two  must  be  united  by  placing  M and  N 
side  by  side,  and  driving  them  together.  Even  then,  with- 
out bolting,  they  can  not  be  drawn  apart  lengthwise ;  nor,  if 
the  joints  be  good,  can  they  be  easily  sprung  in  any  direc- 
tion. At  Jf  the  two  pieces  are  shown  as  they  appear  when 
united. 

Fig.  30  represents  the  same  pieces  that  are  shown  in  Fig. 
29 ;  but  here  drawn  in  a  different  position,  being  placed 
side  wise  toward  the  spectator,  instead  of  endwise  toward 
him  as  in  Fig.  29.  Observe  that  the  measures,  according  to 
the  scale,  are  the  same  in  the  one  case  as  in  the  other.  That 
mode  of  representation  which  will  give  the  best  view  of 
the  object  should  in  all  cases  be  adopted;  and  in  some 
cases  both  modes  should  be  used,  as  the  one  will  often  give 
views  of  parts  that  are  not  shown  in  the  other. 

Fig.  31  shows  a  still  different  mode  of  scarfing,  and  one 
that  is  very  firm,  and  much  more  easily  executed  than  Fig. 
30.  When  the  pieces  are  firmly  bolted  together,  the  inter- 


04  INDUSTRIAL   DRAWING.  [BOOK   NO.  II. 

locking  gives  them  great  additional  strength  against  any 
force  that  might  tend  to  wrench  them  sideways.* 

PROBLEMS   FOR   PRACTICE. 

1 .  Draw  A  of  Fig.  24  with  the  narrow  side  toward  the  spectator. 

2.  Draw  the  two  parts  of  Fig.  25  with  the  narrow  sides  toward  the  spec- 
tator. 

3.  Draw  Tig.  28  with  the  dark  sides  of  both  pieces  toward  the  spectator. 

4.  Design  a  figure  similar  to  Fig.  30,  but  of  different  proportions  in  the 
interlocking  parts. 

5.  What  are  the  contents  of  A,  Fig.  24  ;  the  scale  being  three  inches  to  a 
space  ? 

6.  What  are  the  contents  of  the  entire  Figure  B  ? 

Free-hand  Blackboard  Exercises. — Any  or  all  of  the  fig- 
ures on  page  3. 

PAGE  FOUR.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 
This  page  shows  the  different  methods  adopted  by  ma- 
sons and  engineers  in  laying  bricks.  A  brick  of  ordinary 
size  is  two  inches  thick,  four  inches  wide,  and  eight  inches 
long.  Bricks  which  are  laid  lengthwise  across  the  wall, 
with  their  ends  toward  the  face  of  the  wall,  are  called  head- 
ers; and  those  which  are  laid  with  their  lengths  parallel  to 
the  face  of  the  wall,  are  called  stretchers.  The  two  principal 
methods  of  laying  bricks  in  walls,  where  much  strength 
is  required,  are  what  is  called  the  English  bond  and  the 
Flemish  bond.  In  the  former,  the  face  of  the  wall  always 
shows  the  headers  and  stretchers  in  alternate  layers,  or 
courses,  as  they  are  called  by  bricklayers.  Thus  at  A  is  a 

*  It  may  here  be  remarked  that  the  only  lines  in  Diagonal  Cabinet  Per- 
spective which  are  susceptible  of  direct  measurement  are  the  horizontal, 
vertical,  and  diagonal  lines ;  and  that  when  any  other  line  occurs,  if  it  is  to 
be  measured,  it  must  be  done  by  mathematical  calculation,  after  first  re- 
solving it  into  the  hypothenuse  of  a  right-angled  triangle.  Thus  at  G  the 
lines  2  1  and  1  4  are  measurable ;  the  former,  according  to  the  scale  here 
adopted,  measures  nine  inches,  and  the  latter  thirty-six  inches ;  and  the  an- 
gle 2  1  4  represents  a  right  angle.  Hence  the  line  2  4  is  the  hypothenuse 
of  a  right-angled  triangle ;  and  the  teacher  may  show  the  advanced  pupil 
how  to  ascertain  its  precise  length  by  adding  together  the  squares  of  the 
sides  2  1  and  1  4,  nnd  then  extracting  their  square  root.  And  so  in  all 
similar  cases  of  the  measurement  of  lines  that  are  neither  diagonal,  vertical, 
nor  horizontal. 


CABINET   PERSPECTIVE— PLANE    SOLIDS.  95 

course  of  headers,  and  at  B  is  a  course  of  stretchers ;  and 
in  English  bond  the  face  of  the  wall  shows  alternate  courses 
of  the  two.  For  Flemish  bond,  however,  the  face  of  each 
course  is  composed  of  a  combination  of  headers  and  stretch- 
ers. In  both,  a  brick  of  half  the  ordinary  width,  and  also 
one  of  half  the  ordinary  length,  is  often  required ;  and  these 
half  bricks  should  be  made  in  every  brick-yard. 

Fig.  32  represents  a  wall  eight  inches  thick,  laid  in  En- 
glish bond.  Observe  that  the  first  or  upper  course  is  coin- 
posed  of  headers ;  the  second  of  stretchers ;  the  third  of 
headers ;  the  fourth  of  stretchers,  etc.  In  drawing  these  ex- 
amples (as  in  nearly  all  other  cases  of  drawing),  it  is  neces- 
sary to  draw  the  upper  portions  first — that  is,  to  begin  at 
the  top. 

Fig.  33  represents  a  twelve-inch  wall  in  English  bond. 
Observe  that  the  face  of  the  wall  shows  alternate  courses 
of  headers  and  stretchers ;  while  each  course  is  composed 
of  both  kinds,  arranged  in  a  peculiar  order,  as  shown  in  the 
drawing. 

Fig.  34  represents  a  sixteen- inch  wall  in  English  bond. 
The  face  is  the  same  as  in  the  eight  and  twelve  inch  walls. 
The  upper  two  courses  show  the  arrangement. 

Fig.  35  represents  an  eight-inch  wall  in  Flemish  bond; 
showing  that  each  course  is  composed  of  both  headers  and 
stretchers. 

Fig.  36  represents  a  twelve-inch  wall  in  Flemish  bond. 
Here  half  bricks  of  both  kinds  are  used. 

Fig.  37  represents  a  sixteen-inch  wall  in  Flemish  bond. 

Fig.  38  represents  what  is  called  English  cross  bond. 
Here  the  face  of  the  wall  shows  an  arrangement  in  the  sec- 
ond stretcher  line  by  which  its  joints  come  exactly  below 
the  middle  of  the  bricks  in  the  first  stretcher  line ;  and  the 
same  arrangement  of  stretchers  comes  back  every  fifth  line. 
By  comparing  Fig.  38  with  Fig.  32  the  difference  will  be 
manifest. 

Fig.  39  represents  a  twenty-four-inch  wall  laid  in  Flemish 
bond,  on  the  farther  half  of  which  rises  a  twelve-inch  wall  in 
English  bond,  both  halves  being  capped  with  twelve-by- 
twelve-inch  stones  two  inches  thick.  There  are  also  two 


96  INDUSTRIAL    DRAWING.  [BOOK   ]STO.  II. 

triangular  abutments  resting  on  the  front  half  of  the  lower 
wall,  and  abutting  against  the  upper  wall.  These  abut- 
ments are  twelve  inches  high  at  the  back,  and  their  bases 
project  forward  twelve  inches,  the  width  of  the  lower  half 
of  the  wall.  Their  faces,  therefore,  are  laid  at  an  angle  of 
forty-five  degrees ;  and  hence  each  layer  of  stones,  of  which 
the  abutment  is  composed,  although  only  two  inches  thick, 
presents  a  much  greater  front  surface  view.  The  teacher 
may  require  of  the  advanced  pupil  the  length  of  the  face  of 
the  abutment,  as  designated  by  the  line  2  3  or  4  5.  See 
Note,  page  94. 

PROBLEMS   FOR   PRACTICE. 

1.  Draw  an  eight-inch  wall  in  English  bond,  forty  inches  in  length  at  the 
base,  and  showing  three  partial  courses  and  four  full  courses. 

2.  Draw  a  twelve-inch  wall  in  English  bond,  forty-eight  inches  in  length 
at  the  base,  and  showing  four  partial  courses  and  four  full  courses. 

3.  Draw  a  twelve-inch  wall  in  Flemish  bond,  fifty-six  inches  in  length  at 
the  base,  and  showing  four  partial  courses  and  four  full  courses. 

4.  Let  the  pupil  design  a  twenty-inch  wall  in  English  bond,  and  also  a 
twenty-inch  wall  in  Flemish  bond,  and  so  that  the  bricks  will  break  joint  in 
the  best  manner  possible. 

JBlachboard  ^Exercises. — Any  of  the  figures  of  page  4, 
showing  the  true  size  of  the  bricks. 

PAGE  FIVE.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

Fig.  40  represents  a  cross-shaped  figure  formed  by  so  cut- 
ting a  cube  twenty  inches  square,  that  vertical  sections, 
eight  inches  square  at  the  ends,  are  taken  away  from  its 
four  corners.  The  form  and  extent  of  the  entire  cube  are 
shown  by  the  dotted  lines.  Let  the  pupil  calculate  the 
number  of  cubic  inches  taken  away,  and  the  number  re- 
maining. 

Fig.  41  represents  a  cube  of  the  same  original  size  as  Fig. 
40,  from  which  several  sections  are  cut  away,  leaving  what 
is  shown  in  the  figure.  Let  the  pupil  calculate  the  number 
of  cubic  inches  taken  away,  and  the  number  remaining. 

Fig.  42  shows,  on  its  front  vertical  face,  a  single  fret  of 
the  same  form  as  is  seen  at  Fig.  24,  page  6,  of  Book  No.  I.; 
but  here,  in  Fig.  42,  the  fret  is  shown,  not  as  a  surface  deco- 
ration merely,  but  in  the  solid  form,  four  inches  in  thickness. 


CABINET   PERSPECTIVE — PLANE    SOLIDS.  97 

Observe  that  the  thickness  is  represented  by  one  diagonal 
space. 

Fig.  43  shows  the  same  solid  fret,  but  with  the  addition 
of  a  solid  piece  four  inches  square  running  through  it  cen- 
trally, the  two  being  framed  together. 

Fig.  44  represents  a  frame-work  thirty-four  inches  wide 
and  fifty-six  inches  long,  the  side  pieces  formed  of  timbers 
six  inches  square  at  the  ends,  and  the  end  pieces  formed  of 
timbers  six  by  sixteen  inches  at  the  ends,  the  end  pieces 
being  mortised  into  the  side  pieces  by  tenons  two  inches 
thick,  eight  inches  wide,  and  six  inches  in  length.  Hence 
the  mortises  which  receive  the  tenons  must  extend  entirely 
through  the  timbers.  At  A,  B^  C,  and  D,  the  four  timbers 
which  compose  the  frame  are  drawn  separately,  and  in  their 
relative  positions ;  and  at  F  they  are  put  together.  The 
side  pieces  show  mortises  for  upright  posts.  At  E,  one  of 
the  end  timbers  is  represented  as  standing  vertically  on  one 
of  its  tenons,  with  its  broadest  side  fronting  the  spectator. 
Observe  that,  according  to  the  scale,  it  measures  precisely 
the  same  as  C  or  D. 

PROBLEMS    FOR    PRACTICE. 

1.  Draw  the  lightly  dotted  outlines  of  a  cube  of  the  same  dimensions  as 
Fig.  40 ;  and  then  represent  the  cube  as  it  would  be  after  taking  vertical 
sections  from  its  four  corners,  anl  from  the  centres  of  its  four  bides,  each 
four  inches  square  at  the  ends.     First  mark  out,  on  the  upper  face  of  the 
dotted  cube,  the  upper  face  as  it  would  appear  after  the  pieces  were  taken 
out.     What  would  be  the  solid  contents  of  the  eight  pieces  thus  taken  out  ? 
The  solid  contents  remaining  in  the  cube  ? 

2.  Draw  a  fret  similar  to  Fig.  34,  page  G,  of  Book  No.  L,  representing  the 
solid  form  of  the  fret,  two  inches  in  thickness  ;  but  draw  it  on  an  enlarged 
plan,  so  that  the  width  of  the  fret-line  shall  occupy  a  full  space.     Scale,  two 
inches  to  a  space. 

3.  Draw  F  of  Fig.  44  so  that  the  longest  side  shall  front  the  spectator. 

4.  Draw  the  same  frame  standing  on  one  end,  with  the  broad  side  front- 
ing the  spectator. 

5.  Draw  the  same  frame  standing  on  one  end,  with  the  edye  of  the  frame 
fronting  the  spectator. 

Blackboard  Zeroises. — Any  of  the  figures  and  problems 
on  page  5. 

E 


98  INDUSTRIAL   DKAWING.  [BOOK   NO.  II. 

PAGE  SIX.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

Figs.  45  and  46  represent  the  same  frame-work,  but  placed 
in  different  positions  in  relation  to  the  spectator ;  and  Fig. 
47  represents  these  two  frames  united  by  string-pieces,  in 
the  form  of  a  bedstead.  Let  the  pupil  give  the  dimensions, 
according  to  the  scale  adopted  for  the  page — two  inches  to 
a  space. 

Fig.  48  represents  a  vertical  pillar  twelve  inches  square, 
surrounded  at  the  top  by  a  moulding  four  inches  in  height 
and  projecting  four  inches.  Observe  that  the  projection  of 
the  moulding,  whether  measured  horizontally  by  the  dis- 
tances 1  2  and  5  6,  or  diagonally  by  the  distances  2  3  and 
4  5,  etc.,  is  in  all  parts  four  inches.  The  height  of  the  pillar 
is  the  distance  2  9,  which  is  twenty  inches.  Observe  the 
following  rules:  1.  A  horizontal  moulding,  or  any  hori- 
zontal projection,  conceals  from  view  a  portion  of  the  front 
vertical  surface  immediately  below  it,  equal  in  height  to  one 
half  the  projection.  Thus  the  dotted  line  7  8  indicates  the 
lower  line  of  the  attachment  of  the  moulding  to  the  front 
face  of  the  pillar;  by  which  it  is  seen  that  the  moulding,  four 
inches  in  projection,  conceals  from  view  the  two  inches  in 
height  of  the  front  face  of  the  pillar  immediately  below  the 
attachment  of  the  moulding.  This  principle  is  also  seen  in 
the  fact  that  the  distance  from  the  base  of  the  pillar  to  the 
moulding  is  measured  from  9  to  7 — thus  making  the  mould- 
ing sixteen  inches  above  the  base.  2.  A  horizontal  projec- 
tion conceals  from  view  a  portion  of  tJie  DIAGONAL  vertical 
surface  immediately  beloio  it,  equal  in  height  to  the  extent  of 
the  projection. 

Thus  8  12  is  the  line  of  the  attachment  of  the  under  side 
of  the  moulding  to  the  side  of  the  pillar;  and  12  13  shows 
the  vertical  extent  of  the  side  of  the  pillar  which  the  mould- 
ing conceals. 

But  suppose  the  front  moulding  did  not  project  at  all 
beyond  the  face  of  the  left-hand  side  of  the  column.  It  will 
be  seen  that,  in  that  case,  the  front  end  of  the  moulding 
would  terminate  in  the  line  S  10;  thus  having  an  apparent 
projection  of  two  inches  beyond  the  line  7  9,  the  same  as  the 


CABINET   PERSPECTIVE — PLANE    SOLIDS.  99 

right-hand  end  of  the  moulding  would  seem  to  project  to  the 
left  of  the  line  8  11.  All  the  principles  of  the  projections 
of  rectangular  mouldings  may  be  learned  from  this  figure. 

Fig.  49  represents  the  same  pillar,  twelve  inches  square, 
and  the  same  moulding  as  in  Fig.  48 ;  but  in  Fig.  49  the 
moulding  is  placed  six  inches  below  the  top.  Observe  that 
the  measurements  of  the  projection,  and  of  the  parts  con- 
cealed, are  the  same  in  both  figures. 

Fig.  50  represents  the  same  pillar  as  in  the  preceding  two 
figures  ;  but  here  it  is  placed  upon  a  base  four  inches  thick, 
that  projects  beyond  the  four  faces  of  the  pillar  four  inches 
on  all  sides.  Observe  how  the  measurements  are  made 
here  also. 

Fig.  51.  Here  we  have  represented  eighteen  inches  in 
height  of  the  top  of  a  pillar  which  is  twenty  inches  square 
at  the  ends,  with  a  projecting  moulding  of  four  blocks  on  a 
side,  four  inches  apart,  encompassing  the  upper  end  of  the 
pillar.  These  blocks  are  each  four  inches  in  height  and  two 
inches  in  thickness,  and  have  a  projection  of  four  inches. 
In  drawing  them,  first  trace  lightly  a  moulding  around  the 
pillar,  similar  to  the  moulding  of  Fig.  48  ;  then  mark  out,  in 
this  moulding,  the  divisions  which  constitute  the  blocks. 
Observe  that  the  blocks  on  the  two  diagonal  sides  of  the 
pillar  must  necessarily  be  only  half  a  diagonal  in  thick- 
ness to  represent  two  inches.  Notice  the  portion  of  the 
face  of  the  pillar  below  the  blocks  concealed  by  their  pro- 
jection. 

Fig.  52,  which  represents  a  pillar  resting  upon  a  block 
that  projects  four  inches  from  the  face  of  the  pillar,  has  a 
narrow  moulding  one  inch  thick,  with  a  projection  of  two 
inches,  encompassing  the  pillar  near  the  middle  of  its  height. 
How  far  is  this  moulding  below  the  top  of  the  pillar,  and 
how  far  above  the  top  of  the  projecting  base? 

Fig.  53  represents  a  four-legged  table,  placed  bottom  up- 
ward, so  that  its  several  parts  may  be  shown  in  the  best 
manner  possible.  They  could  not  be  seen  so  well  if  it  were 
represented  in  its  natural  position.  Let  the  pupil  describe 
the  size  and  length  of  the  legs — how  they  are  placed  in  re- 
lation to  the  inclosing  frame ;  width  and  thickness  of  this 


100  INDUSTRIAL   DRAWING.  [BOOK    NO.  II. 

frame,  length  of  its  sides,  and  how  joined  at  the  corners ; 
size  of  the  top  of  the  table,  its  projection  beyond  the  frame, 
thickness  and  number  of  the  boards  which  compose  it,  etc. 

Fig.  54  represents  a  diamond  or  lozenge-shaped  figure 
formed  of  two-space  diagonals,  and  having  a  thickness  of 
four  inches.  What  is  its  length  ?  Its  breadth  ?  Its  front- 
face  measure  in  square  inches?  (See  page  58;  No  7.)  Its 
contents  in  cubic  inches  ? 

Fig.  55  is  the  same  figure  as  54,  but  its  length  is  here 
placed  in  a  front  vertical  position. 

PROBLEMS    FOR   PRACTICE. 

1 .  Represent  a  vertical  piece  of  timber  twelve  inches  square  at  the  ends, 
and  twenty-four  inches  in  height,  surrounded  by  a  moulding  even  with  the 
top,  two  inches  in  height,  and  projecting  four  inches;  also  a  like  moulding 
even  with  the  base  of  the  timber.     Be  careful  to  get  the  correct  height  of 
the  timber,  as  portions  of  the  top  and  bottom  are  concealed  by  the  moulding. 

2.  Make  a  drawing  like  Fig.  47,  with  the  exception  that  the  posts  shall 
rest  upon  a  frame  formed  of  stuff  four  inches  thick  and  four  inches  in  width, 
and  that  a  frame  formed  of  stuff  four  inches  in  width  and  one  inch  in  thick- 
ness (or  height)  shall  rest  upon  the  tops  of  the  posts.     Be  careful  to  trace 
the  outlines  very  lightly  at  first,  and  only  mark  them  firmly  when  the  visi- 
ble portions  are  correctly  represented. 

3.  Draw  a  lozenge  whose  front  face  shall  be  like  Fig.  54,  but  whose  diago- 
nal length  shall  be  twelve  inches. 

4.  Draw  the  same  lozenge  twelve  inches  in  diagonal  length,  but  in  the  po- 
sition of  Fig.  55. 

Blackboard  Exercises. — Figures  47,  48,  49,  and  the  fore- 
going problems. 

PAGE  SEVEN.— SCALE  OF  FOUR  INCHES  TO  A  SPACE. 

All  the  figures  on  this  page,  except  Fig.  G4,  contain  some 
lines  that  are  neither  true  diagonals,  verticals,  nor  horizon- 
tals; yet  their  positions  are  as  accurately  defined  as  the 
positions  of  those  lines  that  are  directly  measurable. 

Fig.  56  represents  a  kind  of  heavy  block  table,  thirty-two 
inches  in  height,  having  its  top  surface  thirty-two  inches 
square,  and  resting  on  a  base  thirty-two  inches  square.  The 
base  bevels  upward,  and  the  top  bevels  downward  to  the 
same  extent.  As  the  line  a  b  is  drawn  diagonally  from  the 
corner  «,  thirty-two  inches,  so  must  the  line  c  d,  drawn  from 
the  corresponding  corner  c,  run  diagonally  in  the  same  di- 


CABINET   PERSPECTIYE PLANE    SOLIDS.  101 

rection  ;  and  its  real  length  is  thirty-two  inches,  but  the  far- 
ther eight  inches  are  not  visible.  It  is  best  to  draw  the 
top  first ;  then  the  front  face ;  then  the  diagonal  lines. 

Fig.  57  represents  the  same  block  table,  with  the  excep- 
tion that  the  vertical  section,  eight  inches  in  length,  is  cut 
out  of  the  centres  of  both  sides  of  the  base.  Observe  that 
as  the  distance  if  is  eight  inches,  so  must  the  distance  g  a 
be  eight  inches  also ;  and  that  a  b  must  run  toward  the  point 
c,  which  is  a  point  on  the  left-hand  side  of  the  table  corre- 
sponding to  d.  And  c  and  d  must  be  the  same  distance 
apart  as  n  and  m. 

Fig.  58  is  the  same  as  Fig.  57,  with  the  exception  that  the 
section  cut  out  here  extends  through  the  entire  central  sup- 
port;  and  Fig.  59  shows  the  same  as  Fig.  58,  with  the  top 
removed. 

Figs.  60,  61,  62,  and  63  are  the  same  as  the  figures  directly 
above  them,  with  their  sides  turned  to  the  front.  Observe 
that  Fig.  60  measures  in  all  respects,  according  to  the  scale, 
the  same  as  Fig.  56.  Thus  the  dotted  vertical  lines  in  the 
base  front  of  Fig.  56,  although  seen  in  a  side  view  in  Fig. 
60,  are  in  the  latter,  as  in  the  former,  the  downward  con- 
tinuation of  the  columnar  support,  and  in  both  cases  are 
eight  inches  apart,  eight  inches  in  length  or  height,  and  at 
the  same  distances  from  the  corners  a  and  c.  And  if  the 
length  of  the  line  a  o,  in  Fig.  60,  should  be  calculated  math- 
ematically, it  would  be  found  to  be  of  the  same  length  as 
the  same  line  a  o  in  Fig.  56.  Let  the  pupil  trace  out  the 
like  measurements  in  Figs.  61,62,  and  63,  with  the  figures 
above  them. 

Fig.  64  represents  a  heavy  frame-work  sixteen  inches  in 
height  or  thickness,  and  seven  feet  four  inches  square,  with 
a  vertical  recess  of  twelve  by  forty  inches  in  the  centre  of 
each  of  its  four  sides,  and  a  vertical  opening  forty  inches 
square  through  the  centre  of  the  frame. 

Fig.  65  represents  the  same  frame-work  as  Fig.  64 ;  but  in 
place  of  the  square  opening  in  the  centre,  there  rises,  above 
the  frame-work  in  Fig.  65  a  four-sided  pyramid,  forty  inches 
square  at  the  base,  and  thirty-six  inches  in  vertical  height. 
Observe  that  the  outlines  of  the  base  of  the  pyramid  are 


102  INDUSTRIAL   DRAWING.  [BOOK   NO.  II. 

the  same  as  the  upper  surface  outlines  of  the  square  open- 
ing in  the  centre  of  Fig.  64,  and  are  represented  by:  the 
same  figures,  1,  2,  3,  J±,  in  both  cases.  The  centre  of  the 
base  of  the  pyramid,  it  will  be  seen,  must  be  the  point  9 — 
the  point  of  intersection  of  the  lines  5  6  and  7  8,  which 
connect  the  centres  of  opposite  sides  of  the  base.  The  ver- 
tical height  of  the  pyramid  above  the  base  must  therefore 
be  the  length  of  the  line  9  10,  which  represents  a  line  drawn 
vertically  upward  from  the  centre  of  the  base  to  the  apex 
of  the  pyramid. 

PROBLEMS    FOR   PRACTICE. 

1.  Draw  a  figure  similar  to  Fig.  56,  but  forty-four  inches  square  at  the 
bottom  and  top ;  and  with  the  central  vertical  support  twenty  inches  high 
from  o  to  ar,  and  only  four  inches  in  thickness. 

2.  Draw  the  same  with  the  side  view  brought  in  front. 

3.  Draw  a  figure  whose  base  shall  be  similar  to  that  of  Fig.  65,  but  whose 
extreme  side  measures  from  corner  to  corner  shall  be  six  feet,  and  the 
height  or  thickness  eight  inches ;  the  recesses  on  the  centres  of  the  sides 
sixteen  by  twenty-four  inches  ;  the  base  of  the  pyramid  twenty-four  inches 
square,  and  its  vertical  height  fifty  inches. 

Blackboard  Exercises.  —  Figs.  56,  57,  58,  and  62.  Also 
draw  Fig.  65,  giving  to  the  frame-work  one  half  the  measures 
denoted  on  the  paper;  but  make  the  pyramid  thirty-six 
inches  in  vertical  height.  The  dark  shades  may  be  desig- 
nated by  heavy  vertical  lines,  with  either  blue  or  white 
crayons. 

PAGE  EIGHT.— SCALE  OF  ONE  FOOT  TO  A  SPACE. 

Fig.  66  may  be  supposed  to  represent  a  block  of  stone 
twelve  feet  square,  and  four  feet  in  height  or  thickness. 
Suppose  that  we  wish  to  place,  centrally,  on  the  top  of  this 
block  a  four-sided  pyramid,  eight  feet  square  at  the  base, 
and  fifteen  feet  in  vertical  height  from  the  top  of  the  base. 

Evidently  the  base  of  the  pyramid  will  be,  on  all  sides, 
two  feet  within  the  outlines  of  the  top  of  the  base  on  which 
it  rests.  The  lines  a  #,  b  f,  f  d,  and  d  a,  representing  the 
outlines  of  the  base  of  the  pyramid,  must  therefore  be  drawn 
two  horizontal  spaces  within  the  upper  side  lines  i  m  and 
n  j  of  the  base,  and  one  diagonal  space  within  the  front 
and  back  lines  i  n  and  mj.  The  centre,  c,  of  the  base  of 


CABINET   PERSPECTIVE — PLANE    SOLIDS.  103 

the  pyramid,  must  evidently  be  at  the  intersection  of  the 
central  lines  o  p  and  h  g,  and  these  lines  will  be  sufficient 
to  designate  it ;  but  it  must  also  be  at  the  intersection  of 
the  diagonal  lines  a  f  and  d  b  of  the  base.  The  apex  of 
the  pyramid  must  be  vertical  to  the  point  c. 

Fig.  67.  The  top  of  the  base  of  this  figure  is  the  same  as 
the  top  of  the  base  drawn  in  the  preceding  figure,  and  on 
this  base  the  pyramid,  of  the  same  dimensions  as  that  desig- 
nated for  Fig.  66,  is  placed.  The  dotted  lines  and  the  let- 
ters are  put  in  to  designate  the  same  points  that  are  given 
in  Fig.  66.  Observe  that  the  point  cc,  the  apex  or  vertex  of 
the  pyramid,  is  taken  fifteen  spaces  vertically  above  the 
centre,  c,  of  the  base ;  and  from  x  lines  are  drawn  to  the 
corners  d,  /,  and  b.  The  fourth  corner-line  of  the  pyramid 
extends  from  x  to  a,  but  it  can  not  be  seen  because  it  is  on 
the  side  opposite  to  the  spectator.  The  line  c  x  is  called 
the  axis  of  the  pyramid. 

The  base  of  Fig.  67  is  represented  as  ten  feet  in  height 
and  twelve  feet  square ;  and  it  has  recesses  in  the  centres  of 
its  sides  six  feet  high,  eight  feet  wide,  and  two  feet  in  depth. 

Fig.  68  is  a  wedge-shaped  pyramid,  eight  feet  square  at 
the  base,  and  eleven  feet  in  vertical  height.  Observe  how 
the  edge,  a  by  of  the  pyramid  is  drawn  so  as  to  be  directly 
above  the  central  diagonal  line,  g  A,  of  the  base. 

Fig.  69  is  the  same  as  Fig.  68 ;  but  the  side  view  of  the 
wedge  is  here  brought  in  front  of  the  spectator.  Observe 
that  the  measurements  are  the  same  in  both  cases.  See  the 
height,  a  g,  in  both. 

Fig.  70  has  the  same  base,  eight  feet  square,  and  the  same 
height,  eleven  feet,  as  the  preceding  two  figures;  but  the 
wedge-shaped  pyramid  is  diminished  to  four  feet  at  the  apex. 
If  the  edge  of  the  pyramid  were  the  line  a  b,  the  figure 
would  be  the  same  as  Fig.  68;  but  the  edge  is  diminished 
two  feet  at  each  extremity  by  cutting  off  a  r  and  s  b. 

Fig.  71  is  a  truncated  pyramid— the  top  being  cut  off 
parallel  with  the  base.  Its  vertical  height  is  the  axis  line 
c  d.  The  lines  forming  the  edges  of  the  sides  are  drawn  to- 
ward a  point  in  the  upward  extension  of  the  line  c  d. 

Fig.  72  represents  the  top  of  a  pillar  in  the  obelisk  form — 


104  INDUSTRIAL    DRAWING.  [BOOK    NO.  II. 

the  top  being  cut  off  in  the  form  of  a  flat  pyramid.  The 
vertex,  x,  must  be  in  the  line  of  the  axis  of  the  pyramid. 
If  the  sides  of  the  block  were  vertical,  we  might  suppose  its 
base  to  be  o,  d,i,n,  in  the  figure  below  it;  then  c  x  would 
be  the  central  line  or  axis. 

Fig.  73  represents  an  obelisk,  which  consists  of  an  upper 
pyramidal  part,  D,  called  the  shaft,  and  the  support  or  base 
on  which  it  rests,  called  the  ped'estal.  The  pedestal  is  di- 
vided into  three  parts :  A,  the  base ;  13,  the  die ;  and  C  the 
cornice.  There  is  often  a  recess,  or  sunken  part,  in  the  die, 
which  contains  the  inscription,  etc.  The  vertical  height  of 
the  shaft  of  Fig.  73  is  seen  to  be  twenty-seven  feet,  by  count- 
ing the  spaces  from  n  to  x,  which  line  is  the  axis ;  and  the 
base  of  the  shaft  is  eight  feet  square.  The  entire  vertical 
height  of  the  pedestal — from  the  bottom  of  the  base  to  the 
top  of  the  cornice — will  be  found  to  be  eleven  feet,  as  meas- 
ured from  m,  the  centre  of  the  bottom  of  the  base,  to  n,  the 
centre  of  the  top  of  the  cornice.  The  point  m  must  be  at 
the  intersection  of  the  lines  which  connect  the  opposite  cor- 
ners of  the  bottom  of  the  base.  The  face  of  the  die  is  seven 
and  a  half  feet  in  height  and  eight  feet  wide ;  and  the  recess 
in  its  centre  is  five  and  a  half  feet  in  height,  and  six  feet 
wide,  so  that  the  recess  is  just  one  foot  within  the  edges  of 
the  die.  As  the  cornice  projects  one  foot,  it  conceals  from 
view  just  one  half  of  a,  foot  of  the  upper  part  of  the  front  face 
of  the  die,  and  one  foot  of  the  upper  part  of  the  side  face 
(see  Rules,  page  98) ;  hence  the  upper  line  of  the  recess  on 
the  right  side  of  the  pedestal  coincides  in  view  with  the  bot- 
tom line  of  the  cornice.  The  recess  in  each  of  the  sides  of 
the  shaft  is  also  one  foot  in  depth.  The  side  corner-lines  of 
the  shaft  are  drawn  toward  a  point  directly  above  x,  in  the 
continuation  of  the  axis.  The  upper  extremity  of  the  shaft 
is  in  the  flat  pyramidal  form,  similar  to  Fig.  72,  but  more 
pointed. 

Fig.  74  is  a  Grecian  fret,  the  same  as  Fig.  25  on  page  6  of 
Book  No. I.,  here  changed  into  the  solid  form;  and  Fig.  75 
is  the  same  as  Fig.  26  on  the  same  page  of  Book  No.  I.,  here 
changed  into  the  solid  form.  In  both  cases  the  thickness  of 
the  fret,  diagonally,  is  the  same  as  the  width  of  its  front  face. 


CABINET   PERSPECTIVE — PLANE    SOLIDS.  105 

To  get  the  full  effect  of  the  d IT. wings  on  this  page, 
view  them  as  directed  on  page  50.  Let  the  pupil  view  his 
own  drawings  in  the  same  manner. 

PROBLEMS   FOR   PRACTICE. 

1.  Draw  n  base  and  pyramid  similar  to  Fig.  67,  but  let  the  base  be  a  cube 
ten  feet  square ;  let  the  pyramid  be  placed  one  foot  within  the  outlines  of 
the  top  of  the  base,  and  let  the  vertical  height  of  the  pyramid  be  eighteen 
feet.     Let  the  recesses  of  the  base  be  only  one  foot  deep,  and  a  foot  and  a 
half  from  the  edges  of  the  sides. 

2.  Draw  a  wedge-shaped  pyramid,  similar  to  Fig.  68,  whose  base  shall  be 
ten  feet  square,  and  whose  vertical  height  shall  be  sixteen  feet. 

3.  Draw  an  obelisk  of  the  following  dimensions :  1st.  The  pedestal :  base, 
fourteen  feet  square,  and  three  feet  in  thickness ;  die,  ten  feet  square  at  its 
base,  and  height  twelve  feet ;  recess  in  die  two  feet  from  edges,  and  one  foot 
in  depth  ;  cornice,  a  foot  and  a  half  in  thickness,  and  projection  beyond  the 
face  of  the  die  one  foot.     2d.  The  shaft :  base  of  shaft  eight  feet  square, 
placed  centrally  on  the  cornice ;  vertical  height  thirty  feet,  and  apex  like 
Fig.  72. 

4.  Draw  a  fret  like  Fig.  75,  with  the  exception  that  the  face  of  the  bands, 
as  seen  in  front,  shall  be  only  six  inches  wide,  and  the  bands  eighteen  inches 
apart,  while  the  diagonal  depth  or  thickness  of  the  band  shall  be  two  feet. 

ItlacJcboard  Exercises. — Draw  and  shade  the  frets,  Figs. 
74  and  75,  on  the  supposition  that  the  ruled  lines  on  the 
blackboard  are  one  foot  apart. 

PAGE  NINE.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

Fig.  76  represents  a  post-and-board  fence,  its  length  be- 
ing here  viewed  diagonally,  with  what  is  called  the  inside 
of  the  fence  exposed  to  view,  so  as  to  show  its  construction. 
The  posts  are  eight  inches  square,  and  twenty-six  inches 
above  ground ;  rails  two  by  four  inches,  the  lower  one  let 
into  the  posts,  and  the  upper  one  resting  on  the  posts,  both 
flush  with  the  front  edge  of  the  posts ;  boards  thirty-four 
inches  in  height,  one  inch  thick,  and  eight  inches  wide,  ex- 
cept the  two  end  boards,  which  are  only  four  inches  wide. 
In  the  drawing  the  posts  are  placed,  for  want  of  room  on  the 
paper,  much  nearer  together  than  they  would  be  in  the  real 
fence. 

Fig.  77  represents  the  same  fence  that  is  shown  in  the 
preceding  figure ;  but  here  its  length  is  placed  fronting  the 
spectator,  and  the  end  is  viewed  diagonally.  Let  the  pupil 

E2 


106  INDUSTRIAL   DKAWIXG.  [BOOK    KO.  II. 

test  the  measurements  of  both  figures,  and  see  if  in  all  re- 
spects they  fully  correspond  with  each  other. 

Fig.  78  is  a  somewhat  elaborate  post-and-rail  fence.  Here 
the  posts  are  eight  by  twelve  inches  in  size,  and  thirty-two 
inches  in  height ;  and  the  rails  are  two  by  four  inches,  the 
lower  three  rails  being  let  into  the  posts  four  inches.  Let 
the  pupil  describe  the  construction  in  full — length,  position, 
distances  apart,  etc.,  of  all  the  rails. 

Fig.  79  represents  a  four-pointed  star  cut  out  of  a  plank 
two  inches  in  thickness.  Let  the  pupil  describe  it :  distance 
of  the  points  from  the  centre,  etc. 

PKOBLEMS   FOK   PRACTICE. 

1.  Let  the  pupil  draw  a  fence  similar  to  Fig.  76 :  posts  to  be  six  inches 
square ;  rails  two  by  three  inches  ;  height  of  posts  and  length  of  boards  the 
same  as  in  the  figure. 

2.  Let  him  draw  the  same  in  the  position  represented  in  Fig.  77. 

3.  Let  him  draw  Fig.  78  so  that  its  length  shall  be  viewed  diagonally.     It 
will  be  found  that  some  objects  can  be  best  represented  in  one  position  and 
some  in  another. 

Blackboard  Exercises. — Fig.  76,  and  problem  2. 

PAGE  TEN.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

Fig.  80  is  another  form  of  post-and-rail  fence,  in  which  the 
rails,  in  their  full  size  of  two  by  four  inches,  are  mortised 
through  the  posts — the  top  of  the  rail  being  in  a  horizontal 
position.  Let  the  pupil  describe  the  posts,  rails,  etc.,  in  full. 
The  posts  are  here  drawn  much  nearer  together,  according 
to  the  scale,  than  they  would  be  placed  in  the  real  fence. 

Fig.  81  will  illustrate  the  principles  of  drawing  the  repre- 
sentation of  a  fence  where  the  square  rails  are  let  into  the 
posts  diagonally.  Let  the  upright  timber  here  represent 
the  post,  which,  however,  is  here  drawn,  for  convenience  of 
representation,  only  two  inches  in  thickness,  and  twenty 
inches  in  width.  Suppose  we  wish  to  represent  a  diagonal 
square  mortise  through  this  post,  for  the  purpose  of  receiv- 
ing a  square  rail  of  twelve  inches  diagonal  diameter.  Lay 
off  the  square  1  3  5  7,  of  twelve  inches  to  a  side,  and  square 
with  the  ends  and  sides  of  the  post.  Mark  the  middle  points 
in  the  sides  of  this  square,  and  connect  them,  measuring 
twelve  inches  from  8  to  4,  and  the  same  from  2  to  £,  and  we 


CABINET   PERSPECTIVE — PLANE    SOLIDS.  107 

shall  have  the  diagonal  square  2  Jf.  6  8.  Represent  this 
square  as  cut  through  the  post,  and  we  shall  have  the  ap- 
pearance of  the  diagonal  square  mortise  which  is  to  receive 
the  rail  placed  in  a  diagonal  position.  Observe  that  the  lines 
2  8  and  4  6  are  drawn  in  the  direction  of  three-space  diago- 
nals. Below  we  have  the  appearance  which  would  be  pre- 
sented by  the  rail  passing  through  the  post.  The  rail  is  so 
placed,  with  reference  to  the  eye  of  the  spectator,  that  one 
side  of  it  appears  very  wide ;  while  the  other  side,  being 
seen  very  obliquely,  appears  very  narrow. 

At  A  we  have  represented  the  posts  of  the  same  size  as 
in  Fig.  80,  with  the  square  rail  passing  through  them  diag- 
onally. 

At  B  we  have  represented  the  side  of  the  post  through 
which  the  rail  passes  as  fronting  the  spectator — as  it  would 
be  drawn  if  the  fence  were  viewed  diagonally  lengthwise. 
In  this  case  the  diagonal  square  mortise  would  be  well  rep- 
resented, but  only  one  side  of  the  rail  would  be  seen,  as  in- 
dicated by  the  dotted  representation  of  it.  Hence  the  di- 
agonal lengthwise  representation  of  such  a  fence  would  not 
be  a  good  one. 

Fig.  82  is  still  another  form  of  post-and-rail  fence,  which 
the  pupil  may  describe. 

Fig.  83.  In  this  figure  the  cross-beam,  which  is  supposed  to 
be  designed  to  sustain  a  heavy  weight,  is  supported  by  two 
braces,  which  are  framed  into  the  cross-beam  and  also  into 
the  posts.  The  under  side  of  the  brace  on  the  left  is  seen  so 
obliquely  that  it  exposes  to  view  only  a  very  narrow  sur- 
face. 

Fig.  84  is  the  same  pattern  of  the  quadruple  or  four-band 
fret  that  is  used  for  the  setting  of  an  Arabian  mosaic  in  Fig. 
43,  page  7,  of  Book  No.  I.  Here,  in  Fig.-  84,  the  fret  alone 
is  given,  and  in  the  solid  form.  In  the  lower  part  of  the 
figure  the  half  diagonal  lines  are  marked  in,  to  show  the 
method  of  representing  the  thickness  of  the  fret.  Each  of 
these  lines,  it  will  be  seen,  is  drawn  in  a  diagonal  direction, 
and  the  length  of  half  a  diagonal.  Understanding  this,  the 
whole  figure  is  very  easily  executed  after  the  original  fret 
has  been  drawn. 


108  INDUSTRIAL   DRAWING.  [BOOK   NO.  II. 

PROBLEMS    FOR   PRACTICE. 

1.  Draw  a  post-and-rail  fence  similar  to  Fig.  80,  but  with  the  square  rails 
inserted  into  the  posts  diagonally.     Let  the  posts  be  four  inches  wide  in 
front,  and  twenty  inches  in  diagonal  width  ;  and  let  the  edges  of  the  diago- 
nally inserted  rails  be  two  inches  from  the  edges  of  the  post.     Omit  the  up- 
per rail,  but  show  the  mortises  for  it. 

2.  Draw  Fig.  82  of  the  same  diagonal  thickness  of  stuff  there  represented, 
but  of  only  half  the  front  Avidth.     Outside  dimensions  same  as  in  the  figure. 

Blackboard  Exercises. — Figures  80  and  82,  of  the  same 
real  size  as  described. 

PAGE  ELEVEN.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

Fig.  85  represents  a  section  of  a  plank  picket-fence,  ac- 
cording to  the  designated  scale.  The  pickets  are  made  of 
stuff  four  inches  wide  and  two  inches  thick.  The  bottom 
rail  to  which  they  are  spiked  is  four  inches  by  five  inches  at 
the  ends ;  and  the  upper  rail  is  three  inches  by  four  inches. 
Observe  that  if  the  spikes  are  driven  in  horizontally,  and 
perpendicular  to  the  face  of  the  pickets,  they  must  have  a 
seemingly  upward  diagonal  direction,  as  indicated  by  the 
line  c  a.  Hence,  if  they  are  driven  into  the  central  line  of 
the  rail,  their  heads  must  be  below  that  line,  as  indicated  in 
the  drawing. 

Fig.  86  represents  a  horizontal  box  eight  inches  square  at 
its  two  open  ends,  and  twenty  inches  in  length,  having  its 
ends  framed  into  and  resting  upon  two  vertical  pieces  of 
plank,  each  sixteen  inches  square  and  two  inches  thick. 

Fig.  87  is  the  Grecian  double  fret  represented  in  Fig.  30, 
page  6,  of  Drawing-Book  No.  I. ;  but  here  drawn  to  a  larger 
scale,  and  put  into  the  solid  form.  Observe  that  the  bands 
are  two  inches  in  thickness,  the  same  in  width,  and  the  same 
distance  apart.  Observe,  also,  that  the  short  corner  diag- 
onal lines  are  all  drawn  to  the  centres  of  the  small  squares ; 
and  that  they  thereby  measure  the  required  thickness  of  the 
fret,  and  also  give  the  correct  diagonal  direction  for  the 
solid. 

Fig.  88  is  also  a  Grecian  double  fret,  of  the  kind  seen  on 
page  5,  of  Book  No.  I.,  Fig.  12.  It  is  here  put  into  the  solid 
form,  and  is  used  for  the  bordering  of  a  tablet,  which  is  sup- 
posed to  be  ornamented. 


CABINET    PERSPECTIVE PLANE    SOLIDS.  109 

Fig.  89  represents  a  heavy  plank  chest,  thirty-six  inches 
square  on  the  bottom,  and  twenty-one  inches  in  height  when 
the  lid  is  closed.  In  the  top  of  the  chest  is  placed  a  tray, 
having  in  it  nine  partitions.  This  tray  rises  three  inches 
ahove  the  body  of  the  chest ;  but  its  partitions  are  only  two 
inches  deep ;  and  when  the  lid  shuts  down  it  incloses  within 
it  the  three  inches'  elevation  of  the  tray.  Let  the  pupil  ex- 
amine and  test  all  the  measurements,  and  see  if  the  lid  will 
accurately  fit  over  the  tray,  and  also  be  even  with  the  out- 
side of  the  chest.  What  is  the  thickness  of  the  top  of  the 
lid? 

PROBLEMS   FOR   PRACTICE. 

1.  Draw  the  representation  of  a  board-picket  fence :  rails  the  same  as  in 
ITig.  85,  but  the  pickets  two  inches  wide,  twenty  inches  long,  made  of  stuff 
two  inches  thick,  and  placed  four  inches  apart. 

2.  Draw  the  representation  of  a  chest  similar  to  Fig.  80.     Suppose  it  to 
be  thirty-eight  inches  square ;  the  body  sixteen  inches  high  ;  top  five  inches 
high,  made  of  stuff  two  inches  thick ;  but  the  tray  made  of  one  inch  stuff, 
and  rising  three  inches  above  the  body.     Divide  the  tray  into  sixteen  square 
divisions,  each  eight  inches  square  (including  the  partitions) ;  and  let  the 
partitions  be  three  inches  deep.     Place  the  cover  on  the  farther  side,  oppo- 
site the  front. 

Blackboard  Exercises. — Figures  85  and  87,  and  problem  1. 

PAGE  TWELVE.— SCALE  OF  SIX  INCHES  TO  A  SPACE. 

Fig.  90  represents  a  solid  octagonal  block,  five  feet  in 
length,  with  a  face  diameter,  on  the  line  1  2  or  3  4,  of  three 
feet.  The  octagonal  form  is  the  same  as  that  of  Xo.  13, 
page  2,  of  Book  No.  I.  Observe  that,  in  drawing  the  visible 
sides  of  the  block,  we  draw  diagonal  lines  from  the  points 
5, 1,  6,  4,  and  7,  and  in  all  cases  a  distance  of  five  diagonals, 
representing  five  feet. 

It  will  be  easy  to  find  the  solid  contents,  in  cubic  inches 
or  feet,  of  such  a  block,  after  the  directions  for  measuring 
surfaces  given  in  the  preceding  book.  Thus,  on  the  scale 
of  six  inches  to  a  space,  the  front  face  of  Fig.  90  measures 
six  square  feet ;  and  as  the  length  of  the  block  is  five  feet, 
the  contents  of  the  block  must  be  five  times  six,  or  thirty 
cubic  feet. 

Fig.  91  has  the  same  front  face  as  the  star  figure  contain- 


110  INDUSTRIAL   DRAWING.  [BOOK    NO.  II. 

ed  within  the  No.  13  just  referred  to,  or  the  same  as  the 
star  figure  No.  11  of  Lesson  VI.,  on  the  same  page.  As  the 
face  of  the  star  form,  Fig.  91,  contains  an  area,  according  to 
the  scale,  of  two  square  feet,  and  as  the  length  of  the  solid 
is  six  feet,  the  solid  contents  of  Fig.  91  would  be  twelve 
cubic  feet. 

Fig.  92  is  a  hexagonal,  or  six-sided  solid,  six  feet  in  length, 
its  two  parallel  ends  being  formed  of  hexagons.  The  pupil 
can  now,  doubtless,  easily  calculate  the  cubic  contents  of 
this  figure. 

Fig.  93  is  an  octagonal  figure,  of  the  same  front  outline  as 
No.  10  of  Lesson  VIII.,  page  2,  of  Book  No.  I.  Six  inches 
within  the  series  of  the  outer  front  lines  is  another  series, 
whose  distance  from  the  outer  lines  is  regulated  by  the 
points  -7,  2,  3,  Jh  Both  the  outer  and  the  inner  lines  are 
drawn  in  the  direction  of  three-space  diagonals.  This  com- 
pleted figure  forms  a  hollow  octagonal  tube,  one  foot  in 
length,  whose  sides  are  six  inches  in  thickness,  with  an  ex- 
treme diameter,  both  vertical  and  horizontal,  of  four  feet. 
The  points  5  and  6,  one  diagonal  space  distant  from  4  and  #, 
regulate  the  drawing  of  the  inner  boundary-lines  of  the  far- 
ther end.  These  lines  also  are  drawn  in  the  direction  of 
three-space  diagonals,  and  are  hence  parallel  to  their  corre- 
sponding front  lines. 

Fig.  94  is  a  solid  octagonal  block,  of  four  feet  vertical  and 
horizontal  diameter,  and  six  feet  in  length.  The  pupil,  re- 
ferring back  to  Book  No.  I.,  should  now  be  able  to  calculate 
its  cubical  contents.  Observe  that  the  shading  is  such  as 
most  clearly  to  distinguish  the  several  visible  sides  of  the 
block. 

Fig.  95.  The  outline  of  the  front  face  of  the  block  here 
represented  forms  a  dodecagon,  or  figure  of  twelve  sides,  the 
same  as  is  shown  on  page  2  of  Book  No.  I. ;  while  the  hol- 
low or  opening  through  the  block  is  hexagonal.  It  is  easy 
to  see  how  the  lines  bounding  the  several  sides  of  the  block 
lengthwise,  both  on  the  outside  and  inside,  are  to  be  drawn 
— all  in  the  direction  of  diagonals.  The  outside  lines  must 
all,  of  course,  be  of  equal  length  ;  and  the  lines  forming  the 
boundary  of  the  farther  face  of  the  dodecagon  must  be  par- 


CABINET   PERSPECTIVE PLANE    SOLIDS.  Ill 

allel  to  those  forming  the  boundary  of  the  front  face.  But 
they  are  made  parallel  with  perfect  ease,  because  their  posi- 
tion and  limits  are  definitely  designated  by  the  ruling  of 
the  paper. 

Let  the  pupil,  referring  back  to  the  outlines  of  the  same 
dodecagon  on  page  2  of  Book  Xo.  I.,  calculate  the  solid  con- 
tents of  Fig.  95. 

Fig.  96  has  the  same  front  face  as  Fig.  95,  but  the  solid 
formed  upon  it  is  only  one  foot  in  length.  It  is  thus  drawn 
in  order  to  show  a  portion  of  the  inner  outlines  of  the  far- 
ther end  of  the  block.  What  are  the  cubical  contents  of 
this  figure? 

Fig.  97,  having  the  same  outline  front  form  as  Fig.  90,  is 
drawn  to  inclose  an  area  four  times  as  large  as  Fig.  90. 
(See  ELEMENTARY  PRINCIPLE,  page  54.)  This  figure  is  a 
hollow  octagon,  two  feet  in  length,  and  having  its  sides  six 
inches  in  thickness.  Observe  that  the  front  outline  is  formed 
by  drawing  two-space  diagonals  throughout ;  observe,  also, 
that  the  inner  boundary-lines  of  the  octagon,  at  both  ends, 
are  the  same  in  direction,  but  necessarily  of  less  length. 

Fig.  98  has  the  same  outline  front  form  as  Fig.  93,  but  is 
drawn  to  inclose  an  area  four  times  as  large  as  Fig.  93. 
(See  ELEMENTARY  PRINCIPLE,  page  54.)  It  is  drawn  of 
three-space  diagonals  instead  of  two.  Observe  how  easily 
and  accurately  in  all  these  figures  the  inside  lines  of  the 
front  face  of  the  octagon  are  drawn,  so  as  to  be  exactly 
parallel  to  the  outside  lines.  Thus,  after  taking  the  point 
1,  in  Fig.  98,  the  line  1  7  is  drawn  in  the  direction  of  a 
three-space  diagonal  until  it  intersects  the  diagonal  line  7 10; 
then  the  line  2  7  is  drawn  as  a  three-space  diagonal,  and  it 
will  intersect  the  diagonal  7  10  in  the  point  7.  In  like 
manner  all  the  front  inner-face  lines  are  drawn.  Observe, 
also,  that  the  points  6  and  5  are  designated  by  being  three 
diagonal  spaces  distant  from  3  and  4-  Then  6  12  and  5  12 
are  drawn  in  the  direction  of  three-space  diagonals,  meeting 
in  the  same  point,  12. 

Fig.  99  is  a  frame-work  which  requires  little  explanation. 
Observe  that  the  braces  must  start  at  equal  distances  from 
the  vertical  post ;  and  that  the  distance  1  2  must  be  the 


112  INDUSTRIAL   DRAWING.  [BOOK    NO.   II. 

same  as  3  4*  The  width  of  the  braces,  as  measured  by  the 
spaces  1  6  and  4  5,  must  also  be  the  same ;  although,  as  the 
side  edge  of  the  nearer  brace  is  seen  most  obliquely,  it  ap- 
pears narrower  than  the  farther  brace.  The  pupil  should 
give  the  measurements  throughout,  and  imitate  the  shading 
with  India  ink  and  the  running  dot-line. 

Fig.  100  represents  a  foot-bridge  resting  on  piers  laid  in 
cut  stone,  or  in  brick  three  inches  thick,  six  inches  wide,  and 
twelve  inches  long.  These  piers  are  three  feet  high,  laid 
up  vertically  on  three  sides,  four  feet  by  five  feet  at  the  base, 
and  two  feet  by  five  feet  at  the  top.  The  extreme  width 
of  the  bridge  is  five  feet;  length,  thirteen  feet;  top  of  rail 
from  top  of  pier,  three  feet  nine  inches.  The  timbers  (called 
sleepers)  on  which  the  floor  is  laid  are  twelve  inches  vertical 
height,  and  nine  inches  wide;  the  flooring  is  represented  as 
of  about  t  wo  and  a  half  inch  plank ;  five  of  the  planks  being 
each  two  feet  wide,  and  the  two  end  planks  each  eighteen 
inches  wide.  The  posts  are  six  by  twelve  inches,  cut  out 
so  as  to  let  half  their  thickness  (three  inches)  rest  on  the 
sleepers.  The  planks  come  out  even  with  the  outer  edges 
of  the  posts,  and  hence  they  project  three  inches  beyond 
the  sleepers.  The  top  rails  are  three  by  six  inches. 

Figs.  101  and  102  represent  patterns  of  cubical  blocks, 
such  as  are  sometimes  worked  in  worsteds  of  three  colors. 
We  have  also  seen  a  carpet  of  the  same  pattern,  although  it 
is  a  very  poor  pattern  for  such  a  purpose.  No  further  di- 
rections are  required  for  drawing  or  shading  these  figures. 

Blackboard  Exercises. — Figures  90,  93,  96,  and  99.  Ob- 
serve that  the  farther  braces  of  Fig.  99  run  downward  in 
the  direction  of  diagonals,  and  the  front  braces  in  the  direc- 
tion of  three-space  diagonals. 

PROBLEMS   FOR   PRACTICE. 

1.  Draw  an  octagonal  solid  similar  to  Fig.  90,  but  whose  front  face  is 
only  one  fourth  as  large.     Be  careful  to  draw  the  lines  of  the  front  face  in 
the  direction  of  two-space  diagonals,  and  half  the  length  of  those  in  the 
figure. 

2.  Draw  a  solid  similar  to  Fig.  91,  but  whose  front  face  shall  be  four 
times  as  large. 

3.  Draw  a  solid  similar  to  Fig.  92,  which  shall  contain  four  times  the 


CABINET   PERSPECTIVE PLANE   SOLIDS.  113 

cubic  contents  of  Fig.  92,  and  whose  front  face  shall  contain  four  times  the 
area  of  Fig.  92. 

4.  Draw  a  dodecagon  hollow  body,  the  extreme  outline  of  whose  front 
face  shall  be  the  same  as  Fig.  95  :  let  the  sides  of  the  hollow  body  be  of 
the  same  length,  but  only  half  the  thickness  of  the  sides  of  Fig.  95. 

5.  Draw  a  figure  the  same  as  Fig.  99,  with  the  exception  that  the  side 
shall  front  the  spectator,  and  the  platform  base  shall  be  one  foot  wider  than 
there  represented. 

G.  Draw  Fig.  100  of  the  same  dimensions  as  given,  but  with  the  side 
fronting  the  spectator. 


DRAWING-BOOK    No.  III. 


CABINET  PERSPECTIVE— CURVILINEAR  SOLIDS. 

IT  has  been  observed  that,  in  all  the  representations  of 
objects  given  in  the  preceding  book,  the  side  of  the  object 
fronting  the  spectator  is  drawn  in  its  natural  form  and  pro- 
portions— this  front  side  being  supposed  to  be  in  a  vertical 
position.  For  example,  the  front  side  of  the  cube  at  -K, 
Fig.  1  of  Book  No.  II.,  is  drawn  an  exact  square.  So  in 
curvilinear  solids,  that  side  fronting  the  spectator  is  drawn 
in  its  natural  form,  and  is  supposed  to  be  in  a  vertical  po- 
sition. Therefore,  if  we  wish  to  represent  a  cylinder  in  cab- 
inet perspective,  and  to  draw  it  with  the  end  fronting  the 
spectator,  that  end  must  be  represented  in  its  natural  form 
as  a  perfect  circle. 

PAGE  ONE.— SCALE  OF  ONE  INCH  TO  A  SPACE. 

Fig.  1  is  a  circle  drawn  from  the  centre,  c,  with  a  radius, 
c  a,  of  five  inches.  It  may  therefore  represent  the  end  of 
a  cylinder  of  ten  inches'  diameter. 

Fig.  2.  Here  c  may  be  taken  as  the  centre  of  the  end  of  a 
cylinder  of  ten  inches'  diameter.  A  cylinder  is  a  solid  whose 
bases  or  ends  are  equal  parallel  circles,  at  right  angles  to  the 
axis  of  the  cylinder.  The  axis  is  the  line  passing  through 
the  centre  of  the  cylinder  at  right  angles  to  the  ends  of  the 
cylinder.  The  circumference  of  a  circle  is  often  called  its 
pe-riph'-er-y;  hence,  also,  we  speak  of  the  periphery  of  a  cyl- 
inder. 

In  Fig.  2,  if  we  suppose  the  cylinder  to  be  ten  inches  in 
length,  the  line  c  ct,  drawn  diagonally  from  the  front  centre, 


CABINET   PERSPECTIVE— CURVILINEAR   SOLIDS.  115 

c,  will  represent  the  axis  of  the  cylinder;  and  c  will  be  the 
central  point  of  the  near  end,  and  d  the  central  point  of  the 
farther  end.     If,  therefore,  we  describe  a  circle  from  the 
point  d,  with  a  radius  of  five  inches,  this  circle,  only  one 
half  of  which  could  be  seen  in  the  solid  cylinder,  would  givo 
the  true  form  and  position  of  the  farther  end  of  the  cylinder. 
Connect  the  extreme  edges  of  these  circles  by  the  lines  p  r 
and  in  n,  and  we  have  all  the  visible  outlines  of  the  cyl- 
inder. 

Fig.  3  is  the  same  as  Fig.  2,  with  the  cylinder  shaded, 
so  as  to  represent  it  in  its  solid  form. .  Observe  here  that 

d,  the  same  as  in  the  preceding  figure,  five  diagonal  spaces 
from  c  (ten  inches),  is  the  centre  from  which  the  visible 
boundary  of  the  farther  end  of  the  cylinder  is  described. 
From  points  in  the  line  c  d  are  to  be  described  the  several 
circular  bands  which  encompass  the  cylinder;  and  these 
bands  will  be  the  same  distances  apart  as  are  the  centres 
from  which  they  are  described. 

Fig.  4.  Here  a  longitudinal  right-angled  section,  embrac- 
ing one  quarter  of  the  cylinder,  is  taken  away.  This  shows 
the  position  of  the  axis,  c  d,  better  than  it  can  be  seen  in  the 
preceding  two  figures.  From  the  centre,  d,  are  described, 
with  the  compasses,  the  visible  portions  of  the  farther  end 
of  the  cylinder.  The  sides  are  drawn  in  the  same  manner 
as  in  Fig.  3. 

Fig.  5  represents  a  cylinder  of  the  same  size,  originally, 
as  in  the  preceding  three  figures,  but  having,  first,  one  quar- 
ter cut  away,  as  in  Fig.  4,  and  then,  in  addition,  all  but  the 
lower  left-hand  quarter  of  the  middle  section  is  taken  away. 
Measurements  of  the  several  sections  may  be  made  on  the 
vertical,  horizontal,  and  diagonal  lines  in  the  same  manner 
as  in  all  plane  solids. 

Fig.  6.  Here,  also,  the  cylinder  is  ten  inches  long,  as  meas- 
ured on  the  line  of  its  axis,  c  d.  The  upper  half  of  the  front 
portion,  four  inches  in  length,  is  first  taken  away ;  then  a 
portion  of  the  farther  part,  but  less  than  a  quarter,  is  re- 
moved. 

Fig.  7  is  the  same  as  Fig.  6,  but  shaded  so  as  to  show  the 
several  parts  more  distinctly. 


116  INDUSTRIAL   DRAWING.  [BOOK   NO.  III. 

Fig.  8  represents  three  quarter-sections  of  a  cylinder  five 
inches  in  diameter  and  four  inches  long. 

Fig.  9  represents  a  cylinder  four  inches  in  length  and  ten 
inches  in  diameter,  with  an  opening  four  inches  square  ex- 
tending through  it  longitudinally  and  centrally.  Here  c  is 
the  centre  of  the  front  end  of  the  cylinder ;  and  at  two  di- 
agonal spaces  from  it  will  be  the  centre  from  which  the 
boundary  of  the  farther  end  is  described.  Circles  running 
around  the  cylinder  are  described  by  the  compasses  from 
points  on  the  axis.  Observe  that  the  length  of  the  cylinder 
is  not  only  measurable  on  the  axis,  but  also  on  the  other  di- 
agonal lines  3^56,  etc.,  which  must  all  be  equal — each 
being  the  measure  of  two  diagonals. 

Fig.  10  represents  the  outlines  of  a  hollow  cylinder  four 
inches  in  length  and  ten  inches  in  diameter,  with  walls  one 
inch  in  thickness.  As  c  is  the  centre  of  the  front  end,  we 
first  describe  from  it  the  outer  circle,  or  periphery,  of  the 
cylinder,  with  a  radius  of  five  inches,  and  then  the  inner  cir- 
cle, with  a  radius  of  four  inches,  which  leaves  one  inch  be- 
tween them  for  the  thickness  of  the  walls.  From  r7,  the 
centre  of  the  farther  end,  we  next  describe  two  like  circles, 
or  parts  of  circles,  the  visible  portion  of  the  smaller  circle 
being  1  2  3y  which  is  the  visible  portion  of  the  farther  inside 
boundary. 

Fig.  11  is  the  same  as  Fig.  10  in  outline,  but  fully  shaded. 

Fig.  12  represents  a  cylinder  four  inches  in  length  and 
ten  inches  in  diameter,  with  a  solid  cylinder,  or  axle,  four 
inches  in  diameter  and  twenty  inches  in  length,  passing  cen- 
trally through  it — the  axis  of  the  longer  cylinder  coinciding 
with  the  axis  of  the  shorter  one.  Here  c  d  represents  the 
axis  of  the  short  cylinder,  and  a  b  the  axis  of  the  long  one. 
The  length  of  either,  or  both,  may  be  measured  in  any  one 
of  the  several  ways  mentioned  for  measuring  Fig.  9. 


PEOBLEMS    FOE    PEACTICE. 

1.  Draw  a  solid  cylinder  six  inches  in  diameter  and  twelve  inches  in 
length. 

2.  Draw  the  same,  but  represent  one  quarter  taken  from  it,  similar  to 
Fig.  4. 


CABINET   PERSPECTIVE — CURVILINEAR   SOLIDS.  117 

3.  Draw  the  lower  left-hand  quarter  of  a  cylinder  whose  entire  dimen- 
sions would  be  eight  inches  in  diameter  and  ten  inches  in  length. 

4.  Draw  the  lower  right-hand  quarter  of  a  cylinder  of  the  same  dimen- 
sions. 

5.  Draw  the  lower  horizontal  half  of  a  cylinder  whose  entire  dimensions 
would  be  eight  inches  in  diameter  and  twelve  inches  in  length. 

6.  Draw  a  hollow  cylinder  of  fourteen  inches  in  diameter,  six  inches  in 
length,  and  one-inch  thickness  of  walls. 

7.  Draw  the  lower  half  only  of  a  hollow  cylinder  of  twelve  inches  in  di- 
ameter, 12  inches  in  length,  and  one-inch  thickness  of  wall. 

Blackboard  Exercises. — The  chalk  crayon  compasses  will 
be  required  here.  See  page  49.  Draw  and  shade  all  the 
figures  on  page  1,  and  also  the  problems. 

PAGE  T \YO.-SCALE  OF  OXE  FOOT  TO  A  SPACE. 

Fig.  13  represents  the  outlines  of  a  solid,  four  feet  by  ten 
feet  at  the  base,  and  eleven  feet  entire  height :  but  the  upper 
part  is  a  semicircle.  The  front  semicircular  outline  is  de- 
scribed from  the  point  c,  and  the  farther  one  from  the  point 
#,  distant,  diagonally,  from  c,  four  feet.  Observe  that  the 
side  line,  a  5,  is  drawn  so  as  to  touch  the  semicircles  at  the 
extreme  limit  of  our  view  of  the  semicircular  top ;  and  that 
the  points  a  and  b  are  necessarily  at  the  intersections  of  di- 
agonals, from  c  and  ce,  with  the  semicircles.  Let  it  be  no- 
ticed that,  as  the  measurements  1  £,  3  4,  5  69  7  £,  9  10,  a  b, 
and  c  x,  are  ail  equal  in  the  real  object,  so  they  must  be 
equal  in  the  drawing. 

Fig.  14  is  the  same  as  the  preceding  figure  shaded. 

Fig.  15  is  similar  to  Fig.  14  ;  but  it  rests  on  a  base  which 
projects  laterally  one  foot;  and  the  semicircular  top  also 
projects  laterally  one  foot,  equal  to  the  projection  of  the 
base. 

Fig.  16  is  the  same  as  the  preceding  figure  fully  shaded. 

Fig.  17  represents  the  outlines  of  an  archway,  formed  of 
two  parallel  walls,  each  two  feet  thick,  six  feet  high,  and 
six  feet  wide,  placed  eight  feet  apart,  and  surmounted  by  a 
semicircular  arch  of  the  same  thickness  as  the  walls  on  which 
it  rests. 

Fig.  18  is  the  same  archway  represented  as  reversed  in 
position,  and  placed  arch  downward.  This  shows  the  out- 


118  INDUSTRIAL    DRAWING.  [BOOK   NO.  III. 

lines  of  the  bases  of  the  walls  in  full,  and  a  part  of  the  un- 
der side  of  the  arch,  which  could  not  be  shown  in  the  draw- 
ing above  it.  Figs.  19  and  20  are  the  same  as  1Y  and  18, 
shaded.  In  this  way  an  object  may  be  represented  in  dif- 
ferent positions,  when  it  is  desirable  to  show  as  fully  as  pos- 
sible the  construction  of  its  several  parts. 

Fig.  21  represents  a  semicircular  arch  resting  on  four  pil- 
lars, two  on  a  side.  The  pillars  are  four  feet  square  and 
fourteen  feet  high,  each  pair  resting  on  a  base  one  foot  in 
thickness,  six  feet  wide,  and  ten  feet  in  length,  and  sur- 
mounted by  a  cap  (capital)  of  the  same  dimensions.  The 
width  of  the  archway,  as  measured  from  a  to  b,  is  twelve 
feet ;  and  the  height,  from  c  to  2,  twenty-one  feet.  From  the 
point  x  the  front  semicircles  are  described  ;  and  from  y  the 
semicircles  which  bound  the  farther  end  of  the  arch  are  de- 
scribed. 

The  divisions  of  the  front  of  the  arch,  2,  3,  4,  &•>  etc.,  are 
laid  oif  by  the  compasses;  and  from  the  points  of  division 
straight  lines  are  drawn  toward  the  centre,  a?,  just  as  the 
lines  would  tend  if  the  real  arch  were  laid  with  cut  stone. 
Let  it  be  observed,  also,  that  the  lines  5  7,  6  £,  9  10,  etc.r 
drawn  in  the  direction  of  diagonals,  must  all  be  of  precisely 
the  same  length  in  the  drawing,  according  to  the  scale,  a? 
they  would  be  in  the  real  arch.  In  fine,  all  diagonal,  ver- 
tical, and  horizontal  lines  in  the  drawing  give,  according  to 
the  scale,  perfectly  accurate  measurements  of  the  structure. 
Parts  of  perfect  circles  may  then  be  measured  or  calculated 
with  the  same  accuracy  as  in  the  real  object. 

PROBLEMS    FOR    PRACTICE. 

1.  Draw  a  figure  similar  to  Fig.  1 4,  but  with  a  width  of  eight  feet  in  front, 
depth  of  two  feet,  and  total  vertical  height  of  ten  feet. 

2.  Draw  one  like  Fig.  19  in  all  respects,  except  that  a  central  vertical 
section,  two  feet  in  depth,  is  represented  as  taken  out,  from  right  to  left, 
throughout  the  entire  archway — thus  leaving  the  front  and  the  rear  of 
the  archway  perfect,  and  taking  out  the  third  part  that  lies  between 
them. 

3.  Eepresent  the  same  inverted — that  is,  by  placing  the  arch  downward. 

4.  Draw  a  figure  similar  to  Fig.  21,  but  of  the  following  dimensions: 
Place  two  columns  on  a  side,  each  two  feet  square  and  twelve  feet  high,  ten 
feet  apart  in  front,  and  each  pair  six  feet  apart  diagonally.     Let  each  pair 


CABINET   PERSPECTIVE CURVILINEAR   SOLIDS.  119 

have  a  base  and  cap  similar  to  those  in  the  figure,  and  also  complete  the 
arch  in  a  manner  similar  to  that  in  the  figure. 

Blackboard  Exercises. — Figures  14,  19,  and  20;  also  the 
foregoing  problems. 

PAGE  THREE.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

Fig.  22.  We  have  here  represented  a  pair  of  braces  rest- 
ing upon  a  platform  two  inches  thick,  eight  inches  wide, 
and  twenty-eight  inches  long,  and  so  placed  as  to  be  equally 
inclined  toward  a  central  line,  2  5. 

Fig.  23  represents  a  platform  the  same  as  in  the  preced- 
ing figure,  but  with  curvilinear  braces  placed  upon  it,  and 
equally  inclined  toward  a  central  line,  2  5.  In  this  case, 
the  bases  of  the  braces,  and  the  height  5  2,  being  the  same 
as  in  the  preceding  figure,  we  take  any  point,  as  y  or  c,  from 
which  we  can  describe,  with  the  compasses,  a  curve  that 
will  pass  through  the  points  1  and  2.  This  point,  y  or  c, 
may  be  farther  from  the  left-hand  brace,  or  nearer,  according 
to  the  kind  of  curve  that  we  desire  for  the  brace :  but,  hav- 
ing taken  it,  we  must  describe,  from  the  same  point,  the  cor- 
responding curve  7  9.  Diagonally  from  y  or  c  we  must 
take  the  point  z  or  x,  making  y  z  or  c  x  equal  to  7  8 ;  and 
from  z  or  x  describe  the  curve  3  4-  This  latter  curve  would 
terminate  at  the  point  10,  the  vertex  of  the  farther  face  of 
the  arch,  on  the  under  side.  For  describing  the  curves  of 
the  right-hand  brace,  we  must  take  corresponding  points  at 
the  left  of  the  platform. 

Fig.  24  shows  the  left-hand  brace  of  the  preceding  figure, 
as  drawn  by  itself  alone.  Here  the  point  10  is  visible,  and 
is  seen  to  be  in  continuation  of  the  curve  3  4> 

Fig.  25.  This  figure  represents  a  semicircular  arch  eight 
inches  (y  i)  in  width,  two  inches'  thickness  of  wall,  described 
with  an  extreme  radius  (c  d  or  y  g)  of  twenty-two  inches, 
and  resting  upon  a  base  eight  inches  wide,  two  inches  thick, 
and  fifty-two  inches  in  length — the  whole  divided  vertically 
in  the  drawing,  and  at  right  angles  to  the  face  of  the  arch,  so 
as  to  separate  arch  and  platform  into  two  equal  halves,  six 
inches  apart.  Suppose  the  two  halves  to  be  brought  togeth- 
er centrally :  the  arch  will  thea  form  a  perfect  semicircle. 


120  INDUSTRIAL   DRAWING.  [BOOK   NO.  III. 

Here  all  the  curvilinear  lines  on  the  left  are  described 
from  the  points  c  and  JB,  and  those  on  the  right  from  the 
points  y  and  z.  Let  the  pupil  observe,  that  while  the  line 
from  10  to  b  is  a  curve  struck  by  the  compasses  from  the 
point  a;,  that  b  a  is  a  straight  line,  drawn  according  to  the 
principles  that  were  explained  in  relation  to  Figs.  13  and 
14.  When  the  curve  from  10  to  b  has  reached  the  point  £, 
its  continuation  is  below  the  visible  surface  of  the  arch;  and 
the  straight  line  b  a  is  the  surface  of  the  arch,  just  as  the 
line  b  a  in  Fig.  13  is  the  surface  of  the  arch  there  repre- 
sented. 

Fig.  26  is  the  same  as  the  preceding  figure  inverted.  The 
upper  sides  of  the  platform  and  arch  here  become  the  under 
sides,  and  the  points  x  c  and  y  2,  consequently,  follow  the  in- 
version. 

Fig.  27  represents  a  hollow  cylinder  of  forty-eight  inches' 
outside  diameter,  as  measured  from  1  to  2 ;  eight  inches  in 
length,  as  measured  from  2  to  $,  4  to  5,  6  to  7,  or  any  where, 
diagonally,  on  the  circumference  ;  and  walls  of  two  inches' 
thickness.  The  two  circles  of  the  front  end  of  the  cylinder 
are  described  from  the  point  t;  and  the  visible  parts  of  the 
two  circles  which  bound  the  farther  end  of  the  cylinder  are 
described  from  the  point  a?,  which  must  be  two  diagonal 
spaces  from  £,  in  order  to  make  the  length  of  the  cylinder 
eight  inches.  The  two  front  circles  should  be  described 
first. 

Fig.  28  is  what  is  called  a  quarter/oil  in  architecture; 
which  is  an  ornamental  figure  disposed  in  four  segments  of 
circles,  the  front  face  of  which  is  a  conventional  resemblance 
of  an  expanded  flower  of  four  petals.  (See  Fig.  36,  page  81.) 
The  front  segments  of  circles,  forming  the  face  of  the  orna- 
ment, are  described  from  the  points  a,  b,  c,  c?,  the  four  angles 
of  a  diagonal  square;  and  as  the  ornament  has  a  depth  of 
eight  inches,  the  segments  of  circles  forming  the  farther 
side  of  the  figure  are  described  from  the  points  ic9  x,  y,  z,  two 
diagonal  spaces  from  the  other  centres.  The  shading  lines 
of  the  segments  of  circles  are  described  from  points  between 
the  front  and  rear  centre?,  on  the  dotted  lines.  In  drawing 
the  figure,  first  describe  the  front  segments  from  the  points 


CABINET   PERSPECTIVE  —  CURVILINEAR   SOLIDS.  121 

«,  &,  c,  c7,  then  from  the  other  points  describe  such  portions 
of  the  other  segments  as  could  be  seen. 

PROBLEMS   FOR   PRACTICE. 

1.  Draw  a  figure  similar  to  Fig.  22,  but  make  the  plank  base  four  inches 
longer,  and  place  the  braces  four  inches  farther  apart,  while  they  shall  be 
of  the  same  height,  5  2.      Then  on  the  lower  side  of  the  plank  base  place 
a  like  pair  of  braces,  same  distances  apart,  etc.,  and  meeting  at  a  point  the 
same  distance  below  the  plank  that  2  is  above  it. 

2.  Draw  Fig.  23  with  changes  similar  to  those  that  were  required  to  be 
made  in  Fig.  22  for  the  preceding  problem,  putting  a  pair  of  curvilinear 
braces  below  as  well  as  above  the  platform. 

3.  Draw  Figs.  25  and  26  united  in  one  figure,  and  on  the  same  base, 
but  on  opposite  sides  of  it,  and  without  any  separation  in  the  base  or  in 
the  semicircular  segments. 

4.  Draw  a  cylinder  similar  to  Fig.  27,  but  with  an  extreme  diameter  of 
forty  inches ;  walls  four  inches  in  thickness,  and  length  of  cylinder  sixteen 
inches. 

5.  Draw  a  quarterfoil  similar  to  Fig.  28 ;  but  take  the  centres  of  the 
front  segments  in  the  four  comers  of  a  diagonal  square  of  only  four  diag- 
onal spaces  to  a  side :   draw  the  outer  segments  with  a  radius  of  eight 
inches,  and  the  inner  with  a  radius  of  six  inches ;  and  make  the  quarterfoil 
only  four  inches  in  diagonal  thickness.     The  measure  for  the  radius  must 
be  taken  on  vertical  or  horizontal  lines. 

After  drawing  the  foregoing  problems,  examine  them  carefully  through 
the  opening  formed  by  the  partially  closed  hand. 

Blackboard  Exercises. — Figs.  22,  23,  25;  and  any  of  the 
problems. 

PAGE  FOUR.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 
Fig.  29  represents  a  bracket  of  forty-four  inches'  height, 
thirty  inches'  projection,  and  eight  inches'  thickness.  The 
front  of  the  bracket  (here  the  diagonal  view)  embraces  two 
segments  of  circles;  the  upper  one  described  with  a  radius, 
3  a  or  3  b,  of  twenty-two  inches ;  and  the  lower  one  with 
a  radius,  1  m  or  1  rc,  of  six  inches.  Observe  that  from  the 
point  3  the  curves  a  b  and/.?  are  described;  and  that  from 
the  point  4  the  curve  c  d  is  described.  The  parts  cp  and 
a  n  are  not  portions  of  the  curves,  but  are  straight  lines. 
From  the  point  1  the  curve  m  n  is  described ;  and  from  the 
point  2  is  described  the  hidden  curve  op.  In  selecting  the 
points  1  2  and  3  4  we  are  influenced  wholly  by  the  kind 

F 


122  INDUSTRIAL   DBA  WING.  j_BOOK   NO.  III. 

and  extent  of  the  curve  which  we  wish  to  describe,  just  as 
in  describing  the  curve  of  the  real  bracket;  but  the  diagonal 
distance  from  1  to  2  and  from  8  to  4  is  determined  by  the 
thickness  of  the  bracket. 

Fig.  30  is  the  same  as  Fig.  29,  shaded  in  full. 

Fig.  31  is  a  bracket  of  precisely  the  same  dimensions  as 
Fig.  30,  but  drawn  in  an  inverted  position,  with  the  top  of 
the  bracket  downward,  by  which  means  the  hollows,  or 
curved  portions,  are  represented  more  fully  than  in  the 
preceding  figure. 

Let  the  pupil  observe  that,  wherever  the  thickness  of  the 
bracket  is  measured,  it  measures  in  all  parts  the  same. 
Thus  the  diagonal  lines  a  c,  d  b,g  h,p  n,  o  m,  s  r,  etc.,  all 
measure  the  same. 

Fig.  32  shows  the  outlines  and  mode  of  drawing  a  plain 
and  regular  trefoil,  which  is  an  architectural  ornament  dis- 
posed in  three  segments  of  circles,  being  a  conventional 
representation  of  three-leafed  clover — which  is,  botanically, 
a  trefoil  plant. 

Observe  that  the  points  -?,#,#,  which  are  the  centres  from 
which  the  front  curves  are  drawn,  are  at  the  three  corners 
of  an  equilateral  triangle ;  and  that  the  points  4,  &,  6,  from 
which  the  visible  portions  of  the  curves  of  the  farther  side 
are  drawn,  are  also  at  the  corners  of  an  equilateral  triangle, 
one  diagonal  space  distant  from  the  front  corners. 

Fig.  33  shows  the  same  trefoil  as  Fig.  32,  but  it  is  here 
cut  into  and  through  a  cylindrical  plank  four  inches  in 
thickness,  and  thirty-six  inches  in  diameter.  Observe  that 
a  and  b  are  the  central  points  for  describing  the  outlines  of 
the  cylinder. 

Fig.  34.  Here  we  have  a  quarterfoil  of  the  same  dimen- 
sions as  in  Fig.  28,  but  here  placed  within  a  cylinder  of  the 
same  dimensions  as  that  shown  in  Fig.  27.  Observe  that 
the  central  point  of  the  frOnt  end  of  the  cylinder  is  tj  and 
that  from  this  point,  with  a  radius  of  eleven  inches,  we  de- 
scribe a  circle  which  touches  the  convex  front  edges  of  the 
quarterfoil;  from  the  same  point  t,  but  with  a  radius  two 
inches  greater,  we  describe  another  circle ;  and  thus  get 
the  front  face  of  the  cylinder.  In  this  manner  the  inclosing 


CABINET   PERSPECTIVE — CURVILINEAR   SOLIDS.  123 

cylinder  should  first  be  drawn;  then  the  four  central  points, 
a,  £,  <?,  d,  of  the  front  face  of  the  quarterfoil,  should  be  desig- 
nated at  the  four  corners  of  a  diagonal  square.  These  points 
must  be  taken  according  to  the  size  of  the  quarterfoil  that 
can  be  drawn;  but  it  will  be  observed  that  they  are  at 
equal  distances,  vertically  and  horizontally,  from  the  central 
point  t.  The  point  x  is  taken  the  same  as  in  Fig.  27;  and 
the  points  tc,ic,y,2,  the  same  as  in  Fig.  28.  To  get  the  full 
effect  of  this  drawing,  view  it  through  the  opening  of  the 
partially  closed  hand.  (See  page  50.) 

PROBLEMS    FOR   PRACTICE. 

1.  Draw  a  bracket  similar  to  Fig.  30,  but  of  the  following  dimensions: 
height  of  bracket,  forty  inches  ;    projection,  2G  inches ;    thickness,  four 
inches ;  the  form  of  curves,  etc. ,  according  to  judgment. 

2.  Draw  the  same  inverted,  similar  to  Fig.  31. 

3.  Draw  a  trefoil  on  the  basis  of  a  triangle  of  thirty-two  inches  to  a  side; 
make  it  eight  inches  in  diagonal  depth  or  thickness ;  and  let  it  have  walls, 
which  shall  be  two  inches  thick,  similar  to  the  walls  of  the  quarterfoil  in 
Fig.  28. 

Blackboard  Exercises. — The  foregoing  problems. 

PAGE  FIVE.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

Fig.  35  is  a  conventional  leaf  pattern,  drawn  so  as  to 
represent  four  leaves  whose  central  surface  lines  form  the 
diagonals  of  a  square;  then  these  leaves  are  represented  as 
being  four  inches  in  thickness,  the  under  sides  of  the  leaves 
being  of  the  same  form  and  dimensions  as  the  upper  sides. 

First :  the  outline  of  one  side  of  the  upper  or  front  sur- 
face of  one  of  the  leaves,  say  A^  is  drawn  according  to  judg- 
ment. Then  it  will  be  easy,  as  shown  in  Book  No.  I.,  to 
draw  the  other  side  symmetrical  with  it  by  the  aid  of  the 
ruled  paper.  Thus  the  curvilinear  outlines  of  the  leaf^l, 
on  both  sides  of  the  central  line,  are  a  little  more  than  one 
space  from  the  point  a;  nearly  three  spaces,  on  both  sides, 
from  the  point  b;  four  spaces  from  the  point  c;  three  spaces 
from  the  point  c?,  etc.  In  this  manner  the  outlines  of  the 
upper  surfaces  may  all  be  accurately  drawn  by  the  hand 
alone. 

Next :  we  have  to  draw  the  visible  portions  of  the  out- 


124  INDUSTRIAL   DEAWING.  [BOOK    NO.  III. 

lines  of  the  under  or  farther  sides  of  the  leaves.  If  we  take 
any  point,  as  $,  in  the  leaf  A^  the  point  directly  opposite 
to  it,  on  the  farther  side  of  the  leaf,  must  be  just  one  diag- 
onal space  from  it — that  is,  its  opposite  point  must  be  seen 
at  4j  the  point  opposite  5  must  be  seen  at  6 ;  the  point 
opposite  r,  the  end  of  the  upper  side  of  the  leaf,  must  be  at 
jt?,  which  latter  point  is  the  end  of  the  lower  side  of  the 
leaf.  In  this  manner  we  may  designate  as  many  of  the 
visible  points  of  the  lower  edges  of  the  leaf  as  we  may  de- 
sire; and  then  we  have  only  to  draw  the  curves  through 
them  to  have  the  outlines  of  the  under  side  of  the  leaf,  so  far 
as  they  can  be  seen  from  our  point  of  view.  In  the  leaf 
marked  A,  the  extreme  point  of  the  under  side  of  the  leaf  is 
seen  at  p,  one  diagonal  space  from  r;  in  the  leaf  B,  the 
extreme  point  of  the  under  side  of  the  leaf  is  seen  at  m,  one 
diagonal  space  from  «/  in  the  leaf  D,  the  extreme  point 
of  the  lower  side  is  seen  at  7i,  one  diagonal  space  from  g ; 
and  in  the  leaf  (7,  the  extreme  point  of  the  lower  side  of  the 
leaf  is  at  x,  one  diagonal  space  from  o ;  and  therefore  it 
could  not  be  seen  unless  the  leaf  were  transparent.  Thus 
the  pupil  will  see  what  portions  of  the  lower  edges  of  these 
four  leaves  must  be  visible,  and  how  he  must  draw  them ; 
and  if  he  will  remember  that  any  point  on  the  farther  side 
of  a  vertical  object  put  in  cabinet  perspective  must  be  in  a 
diagonal  direction  from  its  corresponding  point  on  the 
nearer  or  front  side,  and  will  regulate  the  distance  on  the 
diagonal  according  to  the  scale  adopted,  he  will  find  little 
difficulty  in  drawing  any  solid  curvilinear  object. 

Fig.  36  is  the  same  as  Fig.  35,  only  shaded  to  show  the 
form  distinctly. 

Fig.  37  is  a  solid  isosceles*  triangle,  eight  inches  in  thick- 
ness, as  represented  by  the  diagonal  lines  1  2,  7  S,  and  9  10. 
Let  it  be  understood  that  the  face,  1  7  P,  of  the  triangle  is  in 
a  vertical  position,  fronting  the  spectator.  This  face,  then, 
presents  to  our  view  its  natural  proportions,  and  is  meas- 
ured in  any  direction  by  the  measurement  given  to  the  ver- 
tical and  horizontal  lines  of  the  ruling — two  inches  to  a  space. 

*  An  isosceles  triangle  is  one  which  has  only  two  sides  equal. 


CABINET   PERSPECTIVE — CURVILINEAR   SOLIDS.  125 

Taking  the  length,  a  7,  with  the  compasses,  and  applying  it 
to  the  ruled  lines,  we  find  that  a  7  is  a  little  more  than  twen- 
ty-eight inches  long ;  and  we  find  in  the  same  manner  that 
the  base,  1  9,  is  about  seventeen  inches  long,  as  nearly  as  it 
can  be  measured  by  the  compasses.*  Let  it  be  noticed  here 
that  the  rule  of  diagonal  measures  does  not  apply  to  any 
lines  whatever  on  the  surface,  1  7  9;  because  1  7  9  is  here 
supposed  to  be  a  vertical  surface  fronting  the  spectator, 
and  not  a  horizontal  surface.  See  Rule,  page  85.  Observe 
that  the  point  on  the  under  side  of  the  triangle,  opposite  to 
7,  is  seen  at  2;  the  point  opposite  to  3  is  seen  at  4  >  the 
point  opposite  to  5  is  seen  at  6;  the  point  opposite  to  7  is 
seen  at  #,  etc. — all  in  the  direction  of  diagonals.  The  farther 
face  of  this  solid  triangle,  being  also  in  a  vertical  position, 
must  measure  the  same  as  the  front  face. 

Fig.  38  is  the  same  in  form  and  measurement  as  the  pre- 
ceding figure,  but  inverted  in  position.  We  here  see  the 
base,  1  2  10  9,  quite  fully,  but  no  portion  of  the  sides. 

Fig.  39  is  a  curvilinear  quadrangular  solid,  supposed  to 
be  placed  with  the  front  side  in  a  vertical  position.  It. 
measures  thirty-four  inches  between  the  opposite  angles — 
as  from  1  to  2  and  from  3  to  4,  and  is  four  inches  in  thick- 
ness. The  curvilinear  sides  of  the  front  surface  are  described 
with  a  radius  of  six  diagonals,  or  about  seventeen  inches, 
from  the  four  angles  of  an  erect  square ;  and  the  curvilinear 
sides  of  the  farther  face  are  described  from  the  four  angles 
of  another  erect  square  of  the  same  size,  whose  corners  are 
one  diagonal  space  from  the  corners  of  the  first-mentioned 
square.  Observe  that  no  part  of  the  lower  left-hand  side  or 
edge  of  the  solid  is  visible ;  but  the  dotted  line,  s  ?,  shows 
its  position  on  the  farther  face  of  the  solid. 

Fig.  40  is  the  same  curvilinear  solid  that  is  represented  in 
the  preceding  figure,  but  the  sides  of  the  front  face  are  here 
described  from  the  four  angles  of  a  diagonal  square.  The 


*  The  exact  lengths  of  the  lines  a  7  and  1  9  can  be  found,  inasmuch  as 
each  forms  the  hypothenuse  of  a  right-angled  triangle,  which  the  ruling  on 
the  paper  will  show.  In  calculating  their  lengths,  it  will  he  found  that  a  7 
is  a  little  more  than  twenty-eight  inches  long,  and  that  1  9  is  a  mere  frac- 
tion less  than  seventeen  inches. 


126  INDUSTRIAL   DRAWING.  [BOOK   NO.  III. 

points  for  describing  the  curves  of  the  farther  edges  are 
f9  A,  #,  and  d,  which  are  each  one  diagonal  space  from  the 
other  points. 

Fig.  41  is  a  solid  semicircular  and  rectangular  architect- 
ural band,  the  front  vertical  face  of  which  is  two  inches  in 
width,  and  the  thickness  or  depth  of  which  is  four  inches. 

Taking  the  curvilinear  section  A,  it  will  be  seen  that  the 
two  curves  of  the  front  face  are  described  from  the  point  #, 
with  a  radius  of  three  and  of  four  inches ;  and  that  the  two 
curves  of  the  farther  face  (a  part  only  of  each  being  visible) 
are  described  from  the  point  4,  one  diagonal  space  from  #, 
with  the  same  radii  as  the  front  curves.  The  points  5  and  6 
are,  in  like  manner,  the  centres  for  describing  the  curves  of 
the  section  J5.  The  points  for  the  other  curves  are  also 
given.  The  front  face  is  shaded  with  a  light  wash  of  India 
ink,  and  the  other  parts  by  darker  washes  and  the  pencil. 

PROBLEMS    FOR   PRACTICE. 

1.  Draw  the  leaf  forms  B,  (7,  and  D  at  short  distances  apart,  but  in  their 
present  relative  positions,  and  give  to  each  a  thickness  of  eight  inches. 
'Shade  them  as  in  Fig.  3G. 

2.  Draw  Fig.  37  of  the  same  dimensions  as  described ;  but  let  the  cen- 
tral line,  a  7,  be  in  a  horizontal  position,  twenty-eight  inches  in  length — 
tue  point  7  being  at  the  right  hand. 

3.  Draw  Fig.  38  of  the  same  dimensions  as  Fig.  37,  but  having  the  cen- 
tral line,  4  5,  m  a  horizontal  position — the  point  5  being  at  the  left  hand. 

4.  Draw  an  erect  square  frame  of  forty  inches  to  a  side,  in  the  clear — that 
is,  having  a  clear  measure  of  forty  inches  on  the  inside ;  let  the  depth  be 
eight  inches,  and  the  thickness  of  the  material  one  inch.    Within  this  frame 
place  a  solid  similar  to  Fig.  39,  having  its  curvilinear  sides  described  from 
the  inner  corners  of  the  frame  with  a  radius  of  twenty  inches ;  and  let  the 
solid  also  be  eight  inches  in  thickness.     Shade  the  frame  and  the  inclosed 
solid  properly. 

Blackboard  Exercises. — Fig.  41,  as  much  of  it  as  can  be 
put  on  the  board,  and  problem  4. 

PAGE  SIX.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 
Fig.  42  represents  five  quarter-sections  of  the  rims  of 
wheels,  the  front  faces  or  edges  of  which  are  all  in  the  same 
vertical  plane,  fronting  the  spectator.     If  the  rims  were  en- 
tire, they  would  represent  what  is  called  a  nest,  lying  one 


CABIXET   PERSPECTIVE — CURVILINEAR   SOLIDS.  127 

within  another.  As  the  front  edges  all  lie  in  the  same 
plane,  they  are  all  described  from  the  same  centre,  a,  each 
rim  being  made  two  inches  in  thickness.  The  farther  edges 
of  the  rim  are  likewise  in  one  and  the  same  plane ;  and  as 
the  rims  are  four  inches  wide,  they  must  be  described  from 
a  point,  £,  one  diagonal  space  from  a.  The  lines  crossing  the 
front  edges  of  the  rims  all  radiate  from  the  point  a. 

Fig.  43  represents  the  side  walls  of  a  beveled  tub,  whose 
vertical  height,  when  placed  on  its  bottom,  is  sixteen  inches ; 
extreme  diameter  across  the  top,  twenty-eight  inches ;  and 
extreme  diameter  across  the  bottom,  forty  inches. 

The  top  of  the  tub  being  supposed  to  be  in  a  vertical  plane 
fronting  the  spectator,  and  c  being  its  central  point,  the 
outer  circle  of  the  top  is  described  from  c,  with  a  radius, 
c  #,  of  fourteen  inches ;  while  the  inner  circle  is  described 
with  the  radius,  c  g,  of  twelve  inches. 

Now,  as  the  tub  is  to  be  sixteen  inches  in  extreme  height 
(or  depth),  we  take  the  point  x,  four  diagonal  spaces  from  c; 
and  the  point  x  is  then  the  central  point  of  the  bottom  of 
the  tub.  Hence,  with  x  as  a  centre,  and  with  a  radius,  x  b, 
of  twenty  inches,  we  describe  a  circle  for  the  outer  circum- 
ference of  the  bottom  of  the  tub,  thus  making  its  diameter 
forty  inches.  With  a  radius  two  inches  less,  we  describe 
from  x  the  circle  for  the  inner  circumference  of  the  bottom 
of  the  tub,  which  gives  us  the  visible  part,  dsf,  of  this  inner 
circumference  of  the  bottom.  The  lines  r  p  and  m  n  are 
drawn  tangent  to  (that  is,  touching)  the  outer  circumfer- 
ence of  the  top  of  the  tub,  and  the  outer  circumference  of 
the  bottom  of  the  tub. 

Suppose,  now,  that  we  wish  to  put  circumference  lines 
around  the  tub,  and  passing  through  certain  points,  £,  4, 
and  6,  of  the  surface.  In  the  same  proportions  that  the  line 
i  k  is  divided  by  the  points  £,  4,  and  6>,  divide  the  axis  line 
c  x  by  the  points  .?,  £,  and  5 ;  then,  with  one  point  of  the 
compasses  in  the  point  1,  and  the  other  extended  to  the 
point  2,  describe  the  first  curve ;  with  one  point  in  <<?,  and 
the  other  in  4->  describe  the  second  curve ;  with  one  point  in 
5,  and  the  other  in  6,  describe  the  third  curve.  The  visible 
portion  of  the  inside  of  the  tub  is  viewed  so  obliquely  that 


128  INDUSTRIAL   DRAWING.  [BOOK   NO.  III. 

its  widest  portion,  t  s,  is  hardly  one  third  of  the  apparent 
extent  of  i  k  /  although  both  are,  in  reality,  of  the  same  ex- 
tent. This  figure  should  be  viewed  from  above,  and  on  the 
right,  through  the  partly  closed  hand,  when  the  effect  will 
be  very  striking. 

Fig.  44  represents  nine  cylinders,  each  fourteen  inches  in 
extreme  diameter,  four  inches  in  depth,  and  walls  one  inch 
in  thickness,  placed  in  three  rows  forming  a  square,  so  that 
the  cylinders  touch  one  another. 

Fig.  45  represents  one  of  the  stones  from  a  carved  mould- 
ing. It  measures  thirty-eight  inches  in  height,  twenty-six 
inches  in  breadth,  and  twenty-four  inches  in  thickness  as 
measured  any  where  diagonally.  In  making  a  working 
drawing  of  such  a  figure,  certain  measurements  can  be  given 
definitely  for  directions  to  the  stone-cutter — as  the  height, 
the  breadth,  the  distance  from  b  to  11,  from  11  to  9,  from  9 
to  d;  the  size  of  the  semicircle  described  from  c,  etc. ;  and  all 
these  measurements  will  appear  accurately  in  the  drawing. 
Then  the  curve,  3  5  7  d,  may  be  drawn  so  as  best  to  please 
the  eye.  Having  this  drawn,  the  corresponding  curve  on  the 
opposite  side  must  be  drawrn,  in  all  its  parts,  at  a  diagonal 
distance  from  the  first  curve  representing  twenty-four  inches. 
Thus  all  the  diagonal  lines,  such  as  5  6,  7  8,  etc.,  that  can 
be  drawn  across  the  face  of  the  carving,  must  measure  pre- 
cisely the  same  as  the  lines  c  x,  3  4,  9  10,  11  12,  etc. ;  for 
all  alike  measure  the  thickness  of  the  stone. 

Fig.  46  is  a  conventional  heart-shaped  solid,  sometimes 
used  in  architectural  ornamentation. 

The  points  a  and  b  being  taken  vertically  above  c  and  d, 
a  suitable  distance,  according  to  judgment,  the  face  of  one 
half  of  the  figure  must  be  drawn  by  the  guidance  of  the  eye 
alone,  but  making  the  width  of  the  face  the  same  through- 
out, except  where  it  varies  slightly  at  or  near  the  two  ex- 
tremities. It  will  then  be  easy  to  make  the  face  of  the  oth- 
er half  symmetrical  with  it,  according  to  directions  given 
in  Book  No.  II.  Then,  if  the  thickness  of  the  ornament  is 
to  be  four  inches,  or  one  diagonal,  the  diagonal  width  of  the 
drawing  is  determined  in  the  same  manner  as  in  the  pre- 
ceding figure. 


CABINET    PERSPECTIVE CUBVILIXEAE    SOLIDS.  129 

PROBLEMS   FOB   PBACTICE. 

1 .  Draw  five  quarter-sections  of  the  rims  of  wheels,  similar  in  all  re- 
spects to  those  in  Fig.  42,  but  placed  in  the  upper  right-hand  quarter  of  the 
circle. 

2.  Draw  the  outlines  of  a  bottomless  tub  similar  to  Fig.  43,  but  of  the  fol- 
lowing dimensions.     Extreme  upper  (front)  diameter,  twenty-four  inches ; 
vertical  height,  measured  on  the  axis  of  the  tub,  twenty-four  inches ;  ex- 
treme diameter  of  bottom  of  tub,  thirty-four  inches ;  thickness  of  walls,  one 
inch. 

Blackboard  Exercises. — The  foregoing  problems. 

PAGE  SEVEN.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

Fig.  47  represents  a  cubical  block,  twenty -eight  inches 
square,  having  semicircular  grooves  of  four  inches'  radius 
running  diagonally  through  the  centres  of  its  upper,  lower, 
and  right  and  left  hand  sides,  with  a  circular  aperture  of 
four  inches'  radius  running  from  the  front  face  diagonally 
through  the  centre  of  the  block. 

Observe  that  the  centres  a,  #,  c,  d,  <?,/",  from  which  the 
semicircles  on  the  front  and  on  the  farther  side  are  de- 
scribed, are  situated  at  the  middle  point  of  the  edges  of  the 
cube,  just  where  they  would  be  placed  for  marking  out  the 
semicircular  grooves  on  the  real  cube. 

Figs.  48  may  be  supposed  to  be  cut  out  of  a  block  twen- 
ty-eight inches  in  height  (m  n  or  e  g) ;  twenty-eight  inches 
in  depth  (m  o  or  e/);  and  thirty-two  inches  in  width  (m  e 
or  n  (/).  The  figure  consists  of  a  frame-work  having  four 
corner  posts  of  two  by  four  inches,  supporting  a  half-cylin- 
der above  and  one  below,  with  their  concave  surfaces  out- 
ward, the  whole  cut  out  of  the  block.  The  thickness  of  the 
walls  of  the  half-cylinders  is  four  inches.  Their  semicircular 
ends  are  described  from  the  points  tf,  £,  c,  and  d.  The  point 
dt  for  describing  the  farther  end  of  the  lower  half-cylinder, 
must  evidently  be  at  the  middle  of  the  farther  lower  edge 
of  the  block,  just  as  c  is  at  the  middle  of  the  front  lower 
edge.  But  c  is  eight  spaces,  horizontally,  to  the  left  of  the 
corner  #/  so  must  d  be  eight  spaces,  horizontally,  to  the  left 
of  the  corner  h.  If  the  pupil  will  carry  out  all  the  meas- 
urements, just  as  they  are  in  the  real  object,  he  will  find 

F2 


130  INDUSTRIAL   DRAWING.  [BOOK    NO.  III. 

but  little  difficulty  in  the  accurate  representation  of  the 
most  intricate  plans  and  patterns. 

Fig.  49  represents  two  cylinders,  lying,  horizontally  and 
diagonally,  side  by  side  on  a  frame-work  of  twenty-four  by 
thirty- two  inches.  As  the  centres  of  the  farther  ends  of 
these  cylinders  are  at  the  points  b  and  c7,  the  length  of  their 
axes,  a  b  and  c  d,  and,  consequently,  the  length  of  the  cylin- 
ders, is  twenty-four  inches.  The  cylinders  are  placed,  at 
both  extremities,  four  inches  from  the  ends  of  the  frame. 
Complete  the  circle  of  two-inch  radius,  of  which  d  is  the 
centre,  and  the  circle  will  strike  the  frame  at  the  same  dis- 
tance from  the  farther  end  that  the  circle  described  from  c 
strikes  it  from  the  front  end. 

Fig.  50.  This  figure  consists  of  two  portions,  A  and  B  • 
A  being  the  same  as  B  inverted. 

Taking,  first,  the  portion  marked  A:  the  two  semicircu- 
lar curves  a  and  #,  of  the  front  face,  are  described  from  their 
common  centre,  y ;  while  the  two  marked  c  and  d,  of  the 
farther  face,  are  described  from  their  common  centre,  z.  In 
a  similar  manner,  the  two  semicircular  curves  a  and  £,  of 
the  front  face  of  B,  are  described  from  w;  and  the  curves 
c  and  d,  of  the  farther  face,  are  described  from  a*. 

Fig.  51  consists  of  a  series  of  the  same  forms  as  A,  of  the 
preceding  figure,  here  framed  together.  The  pupil  should 
now  be  able  to  describe  this  frame-work  in  full,  and  to  des- 
ignate the  respective  centres  from  which  the  several  pairs 
of  curves  (front  curves  and  rear  curves)  are  described. 

Fig.  52  consists  of  a  series  of  the  same  forms  asJB,  of  Fig. 
50,  here  framed  together;  or  the  same  as  Fig.  51  inverted. 
The  pupil  should  now  be  able  to  describe  it  in  full.  In 
making  a  drawing  of  such  figures — either  in  copying,  or  in 
drawing  from  a  description— the  front  portions  should  be 
drawn  first. 

Fig.  53  is  a  combination  of  the  two  figures  51  and  52; 
requiring  merely  the  placing  of  the  two  figures  in  proper 
position,  and  the  completing  of  the  vertical  corner  posts, 
which  are  left  short  in  Figs.  51  and  52. 

In  making  a  drawing  of  Fig.  53,  the  front  face  should  first 
be  drawn,  then  the  right-hand  side;  then  the  other  parts 


CABINET   PERSPECTIVE — CURVTLIXEAR   SOLIDS.  131 

snoald  first  be  traced  lightly,  and  only  filled  in  firmly  when 
the  positions  of  all  the  parts  are  distinctly  seen. 

PROBLEMS   FOE   PRACTICE. 

1.  Draw  a  block  the  same  in  size  as  Fig.  47,  but  with  semicircular  grooves 
of  eight  inches'  radius  taken  out  centrally  and  diagonally  through  four  of 
the  sides,  as  in  Fig.  47. 

2.  Draw  a  figure  similar  to  Fig.  48  ;  but  let  the  height  of  the  front  be 
twenty-eight  inches,  its  width  twenty-four  inches,  and  the  diagonal  depth  of 
the  block  twenty  inches  ;  let  the  greater  curves  of  the  semicircular  grooves 
be  struck  with  a  radius  of  ten  inches,  and  the  lesser  curves  with  a  radius  of 
eight  inches. 

3.  Draw  a  solid  cylinder  of  six  inches'  radius,  and  twenty-four  inches'  di- 
agonal length,  and  let  it  rest  centrally  upon  a  diagonal  platform  twelve 
inches  wide  in  front,  four  inches  thick,  and  twenty-four  inches'  diagonal 
length,  so  that  the  ends  of  the  cylinder  shall  be  even  with  the  ends  of  the 
platform. 

Blackboard  Exercises. — Figs.  48  and  50,  and  problems  1 
and  3. 

PAGE  EIGHT.— SCALE  OF  THREE  INCHES  TO  A  SPACE. 

Fig.  54  represents  a  wheel  eleven  feet  and  a  half  in  ex- 
treme diameter;  rim,  thirty  inches  wide  and  six  inches 
thick ;  the  cylindrical  nave  or  hub,  four  feet  in  diameter 
and  thirty  inches  in  length ;  and  having  eight  flat  spokes 
or  radii,  each  six  inches  thick,  and  the  same  width  as  the 
rim  and  hub. 

The  foregoing  dimensions  being  given,  we  proceed  in  the 
following  manner  to  make  a  drawing  of  the  wheeL 

1st.  Taking  x  as  the  centre  of  the  vertical  front  face  of 
the  wheel,  with  a  radius,  x  <r,  of  five  feet  nine  inches,  de- 
scribe the  circle  abed  for  the  circumference  of  the  wheel, 
being  eleven  feet  and  a  half  in  diameter. 

2d.  Six  inches  within  the  circumference,  with  the  radius 
x  e,  describe  a  circle,  which,  with  the  former,  will  give  the 
thickness  of  the  rim — six  inches. 

3d.  With  the  radius  ie/,  of  two  feet,  describe  a  circle  for  the 
circumference  of  the  hub,  four  feet ;  and  with  the  radius  x  g, 
of  one  foot,  describe  a  circle  for  the  cylindrical  opening  which 
is  to  receive  the  axle,  which  is  to  be  two  feet  in  diameter. 

4th.  Trace  lightly  the  vertical,  horizontal,  and  diagonal 


132  INDUSTRIAL   DRAWING.  [BOOK    NO.  III. 

lines  through  the  centre,  x,  hereby  dividing  the  rim  into  eight 
equal  parts ;  and  on  each  side  of  these  several  lines  lay  off  a 
space  of  three  inches,  and  draw  the  lines  representing  the 
thickness  of  the  spokes,  terminating  them,  in  one  direction, 
by  the  inner  circle  of  the  rim,  and  in  the  other  direction  by 
the  outer  circle  of  the  hub.  This  completes  the  outlines  of 
the  front  vertical  face  of  the  wheel. 

5th.  As  the  rim  is  to  be  thirty  inches  wideband  the  hub 
and  spokes  of  the  same  diagonal  extent,  we  take  x  y,  thirty 
inches,  for  the  axis,  and  also  for  the  thickness  or  width  of 
the  wheel.  Then  y  will  be  the  centre  of  the  farther  verti- 
cal face  of  the  wheel. 

6th.  From  the  centre,  y,  and  with  the  radius  y  z,  of  five  feet 
nine  inches,  describe  the  outer  circumference  of  the  farther 
face  of  the  wheel;  and  from  the  same  point  y,  with  a  radius 
six  inches  less,  that  is,  with  a  radiuses,  describe  a  circle  (only 
half  of  which,  h  ij,  will  be  visible)  for  the  inside  circumfer- 
ence of  the  farther  face  of  the  rim  ;  then,  with  a  radius,  y  r, 
of  two  feet,  describe  a  circle  (of  which  k  Us  a  part)  for  the 
outer  circumference  of  the  farther  face  of  the  hub;  and 
finally,  with  a  radius,  y  p,  of  one  foot,  describe  a  circle  (of 
which  m  n  is  the  only  visible  part)  for  the  inner  circumfer- 
ence of  the  farther  face  of  the  hub.  All  the  necessary  cir- 
cles will  thus  be  completed — drawn  in  the  very  same  man- 
ner, from  the  two  centres,  #,  y,  of  the  axis,  that  they  would 
be  drawn  from  on  the  real  wheel. 

7th.  The  positions  and  correct  width  of  the  front  faces  of 
the  spokes  having  been  marked  out  under  the  fourth  divi- 
sion, we  have  only  to  draw  from  their  extremities  the  diag- 
onal lines  1  2,  3  4-,  5  6,  7  8,  etc.,  across  the  inside  of  the  rim 
and  the  outside  of  the  hub,  and  the  lines  for  the  farther  vis- 
ible edges  of  the  spokes,  and  the  outlines  of  the  spokes,  so 
far  as  they  are  visible,  will  be  completed. 

8th.  Shade  the  several  parts  in  such  a  manner  as  to  make 
each  part  distinct  from  the  others,  but  at  the  same  time 
paying  as  much  attention  as  you  can,  consistent  with  dis- 
tinctness, to  the  laws  of  light  and  shade.  The  light  in  this 
drawing,  as  in  most  of  the  preceding,  comes  diagonally  from 
the  upper  left-hand  corner. 


CABINET   PERSPECTIVE— CURVILINEAR    SOLIDS.  133 

Observe  with  what  accuracy  all  the  parts,  in  accordance 
with  the  scale  laid  down,  and  with  the  principles  of  repre- 
sentation adopted,  maintain  their  correct  measurements,  the 
same  as  in  the  real  object.  Thus  all  measurements  at  right 
angles  across  the  outside  or  the  inside  of  the  rim  of  the  real 
wheel,  and  across  the  outside  or  the  inside  of  the  hub,  meas- 
ure the  same,  thirty  inches.  So,  also,  in  the  drawing,  all  the 
diagonal  lines  across  the  rim  or  the  hub — as  1  2,3  ^5  6, 7 8, 
v  w,  etc. — measure  precisely  the  same,  five  diagonal  spaces, 
or  thirty  inches. 

Let  this  wheel  be  viewed  as  directed  on  page  50.  The 
pupil  should  not  only  draw  this  wheel,  but  also  one  of  a  dif- 
ferent size,  and  with  the  spokes  arranged  differently. 

PROBLEM:  FOR  PRACTICE. 

1.  Draw  a  wheel,  similar  to  Fig.  54,  of  the  following  dimensions:  Ex- 
treme diameter,  ten  feet  and  a  half;  rim,  thirty-six  inches  wide  and  three 
inches  thick ;  the  cylindrical  nave,  or  hub,  thirty-six  inches  in  diameter, 
three  inches  in  thickness,  and  thirty-six  inches  in  length  ;  and  having  eight 
flat  spokes  or  radii,  each  three  inches  thick,  and  the  same  width  as  the  rim 
and  hub. 

Blackboard  Exercise. — Let  the  pupil  plan  and  draw  a 
wheel  similar  to  that  described  in  the  problem,  and  of  such 
size  as  the  board  will  permit. 

TAGE  NINE.— SCALE  OF  THREE  INCHES  TO  A  SPACE. 

Fig.  55  represents  a  wheel  eleven  feet  and  a  half  in  ex- 
treme diameter;  extreme  width  of  double  rim  and  double 
hub,  four  feet;  thickness  of  rim,  three  inches;  extreme  di- 
ameter of  the  hub,  four  feet,  and  diagonal  extent  of  the  two 
parts,  four  feet ;  thickness  of  rim  of  the  hub,  three  inches ; 
thickness  of  the  broad  paddle-like  spokes,  three  inches,  and 
width  four  feet. 

This  figure  consists  of  two  similar  wheels,  the  second  being 
placed  immediately  back  of  the  first,  and  each  having  its 
separate  rim  and  hub,  but  the  two  being  united  in  one  wheel 
by  the  broad  paddle-like  spokes,  which  are  common  to  both. 

1st.  In  drawing  this  figure,  draw  the  front  wheel  first,  in 
the  same  manner  that  Fig.  54  was  drawn — w  being  the  cen- 
tre of  the  front  face  of  the  wheel,  and  x  the  centre  of  the 


134  INDUSTRIAL   DRAWING.  [BOOK   NO.  III. 

opposite  side.  Draw  the  circles  lightly  with  the  compasses ; 
then  draw  lightly,  in  like  manner,  the  front  and  rear  circles 
of  the  farther  wheel,  from  their  centres  y  and  z. 

2d.  Observe  that  the  eight  broad  spokes  are  arranged  in 
a  manner  different  from  those  in  Fig.  54,  there  being  no 
spokes  on  the  diagonal  lines  3  12  and  7  a ;  and  we  have 
placed  none  on  these  lines,  because,  in  these  positions,  there 
could  be  no  side  view  of  the  spokes  that  are  in  the  direction 
of  3  12,  as  there  is  none  in  Fig.  54.  This  is,  therefore,  a  bet- 
ter arrangement  of  these  portions  of  the  wheel. 

3d.  In  order  to  arrange  the  eight  spokes  equidistant  from 
one  another,  and  intermediate  between  the  centrally  verti- 
cal, horizontal,  and  diagonal  lines  of  the  wheel,  first  draw 
these  lines  lightly,  and  mark  their  points,  as  1, 3, 5,  7, 9, 11, 
etc.,  on  the  circumference  of  the  front  face  of  the  wheel ; 
then  take  the  points  2,  ^,  6, 8, 10,  etc.,  intermediate  between 
them,  and  connect  the  opposite  points  by  light  lines  passing 
through  the  centre,  w.  On  each  side  of  these  latter  lines 
lay  off  half  a  space  (an  inch  and  a  half),  and  draw  lightly 
the  lines  which  determine  the  front  width  of  the  spokes. 
Each  pair  of  these  lines  passes  equidistant  on  both  sides  of 
the  centre,  w.  The  extreme  ends  of  the  spokes  are  then  all 
drawn  diagonally  across  both  parts  of  the  rim,  and  the  oth- 
er ends  of  the  spokes  are  drawn  in  like  manner  across  both 
parts  of  the  hub.  The  dotted  continuations  of  the  lines  show 
where  the  lines  would  appear  if  the  parts  which  obstruct 
the  view  were  transparent. 

Observe  how  the  broad  spokes  are  let  in,  centrally,  be- 
tween the  two  rims  which  compose  the  hub,  and  that  they 
are  let  in  the  exact  depth  of  the  rims,  three  inches.  The 
portions  that  are  let  in  are,  therefore,  continuous  with  the 
inner  sides  of  the  rims  of  the  hub.  Every  part  of  this  wheel 
will  show  beautifully  when  viewed  as  directed  on  page  50. 

If  the  pupil  will  be  careful  to  have  all  the  measures  cor- 
respond with  the  measures  of  the  real  wheel  of  the  given  di- 
mensions, and  will  connect  those  points,  and  those  only,  that 
are  connected  in  the  real  wheel,  he  will  have  no  difficulty 
in  drawing  this,  and  all  similar  representations  of  real  ob- 
jects, with  mathematical  accuracy. 


CABINET   PERSPECTIVE — CURVILINEAR   SOLIDS.  135 

PAGE  TEN.— SCALE  OF  THREE  INCHES  TO  A  SPACE. 

Fig.  56  is  a  drawing  of  what  is  called  a  croicn-wheel, 
which  is  a  wheel  with  cogs  or  teeth  cut  out  of  the  rim,  and 
get  at  right  angles  to  the  plane  of  the  wheel.  See  a  differ- 
ent representation  of  this  same  wheel  in  Fig.  23  of  Book 
No.  IV. 

In  this  figure  the  ends  of  the  projecting  cogs  or  teeth, 
thirty-two  in  number,  form  the  vertical  front  face  of  the 
wheel,  which  has  w  for  its  centre ;  and  the  extreme  diameter 
of  the  wheel — 1  5  or  3  7 — is  eleven  feet.  The  rim,  including 
the  teeth,  is  thirty  inches  wide  and  -three  inches  in  thick- 
ness, and  the  cogs  are  one  foot  in  length ;  the  four  support- 
ing arms  (or  spokes)  of  the  wheel,  which  are  six  by  twelve 
inches,  are  set  back  even  with  the  farther  face  or  side  of  the 
wheel,  and  are  back  eighteen  inches  from  the  ends  of  the 
cogs :  x  is  the  centre  of  the  farther  vertical  side,  and  hence 
to  x,  the  proper  axis,  is  thirty  inches  in  length ;  but  v  x,  the 
axis  of  the  short  hub,  is  only  twelve  inches  in  length.  The 
hub  is  eighteen  inches  in  diameter,  as  measured  from  m  to  w, 
and  its  rim  is  three  inches  in  thickness.  Through  the  hub 
passes  an  axle  on  which  the  wheel  turns,  twelve  inches  in 
diameter  and  sixty  inches  in  length,  having  s  t  for  its  axis. 

1st.  In  drawing  a  figure  with  the  dimensions  given  for 
this  wheel,  first  take  some  point,  as  w,  for  the  centre  of  the 
front  face,  and  with  a  radius,  w  3,  of  five  and  a  half  feet,  de- 
scribe the  outer  front  circumference  of  the  circle  of  cogs ; 
and  then,  with  a  radius  three  inches  less,  describe  the  inner 
front  circle,  which  gives  the  thickness,  three  inches,  of  the 
cogs.  From  x  describe  two  like  circles  for  the  farther  edge 
of  the  rim. 

2d.  Divide  the  front  circumference  into  eight  equal  parts, 
by  the  dotted  vertical,  horizontal,  and  diagonal  lines,  radi- 
ating from  the  centre,  w;  then,  with  the  compasses,  make 
eight  equal  divisions  in  each  of  these  parts,  which  will  give 
to  each  part  four  equal  cogs,  with  spaces  between  them  of 
the  same  width  as  the  cogs. 

3d.  As  the  cogs  are  to  be  twelve  inches  in  length,  take 
the  point  y,  twelve  inches  diagonally  from  «?,  and  with  the 


136  INDUSTRIAL    DRAWING.  [BOOK    NO.   III. 

same  radius  as  w  3,  describe  a  circle;  and  also  describe  one 
with  a  radius  three  inches  less;  and  these  two  circles  will 
give  the  true  measurable  length  of  the  teeth,  both  on  the 
inside  and  on  the  outside  of  the  rim. 

4th.  Now  if,  in  the  real  wheel,  the  cogs  bevel  outward 
from  the  centre,  w,  they  must  be  so  drawn  here,  in  the  follow- 
ing manner:  The  lines  at  the  ends  of  the  cogs  that  limit 
their  width  (not  thickness)  must  all  be  directed  toward 
their  common  centre,  w — as  c  ?o,  a  w,  3  w,  etc. ;  but  the  cor- 
responding lines  at  the  base  of  the  cogs  must  be  directed 
toward  their  common  centre,  y — as  I  y,  9  y,  g  y,  p  y,  etc. — 
just  as  they  are  in  the  real  wheel. 

5th.  All  lines  that  bound  the  teeth  lengthwise — as  h  i,j  I; 
o  r,  etc. — must  be  drawn  in  the  direction  of  diagonals.  These 
lines,  however,  are  drawn  next  after  the  end  lines  that  limit 
the  width  of  the  cogs;  and,  when  drawn,  they  fix  the  posi- 
tion of  the  base  lines  of  the  cogs. 

6th.  From  the  point  v,  the  centre  of  the  front  face  of  the 
short  hub,  describe  the  two  circles,  one  with  a  radius  of  nine 
inches,  and  the  other  with  a  radius  of  six  inches,  for  the 
front  face  of  the  hub ;  and  from  a;,  with  a  radius  of  nine 
inches,  describe  a  circle  (only  a  small  part  of  which  is  seen) 
for  the  outer  circumference  of  the  farther  face  of  the  hub. 
From  s  and  t  are  described  the  circles  for  the  ends  of  the 
axle. 

7th.  As  v  is  the  centre  of  the  vertical  circle  which  limits 
the  front  ends  of  the  spokes,  from  v,  with  a  radius  of  five 
feet  and  three  inches,  describe  a  circle  for  that  purpose.  The 
central  lines  of  the  front  faces  of  the  spokes  must  then  pass 
through  v,  and  the  width  of  the  spokes  is  laid  out  at  equal 
distances  on  each  side  of  these  central  lines.  The  shading 
is  similar  to  that  of  the  preceding  two  figures. 

The  dimensions  of  the  wheel  are  well  seen  in  the  dotted 
squares  AB  CD,  E  F  Q II,  which  bound  its  front  and  rear 
vertical  faces. 

PAGE  ELEVEN.— SCALE  OF  THREE  INCHES  TO  A  SPACE. 

Fig.  57  represents  the  face  or  side  of  a  ratchet  wheel,  a 
wheel  which  has  teeth  like  those  of  a  circular  saw.  The 


CABINET  PERSPECTIVE — CURVILINEAR   SOLIDS.  137 

teeth  may  be  either  long  and  sharp-pointed,  or  short  and 
more  blunt  in  appearance;  but  in  either  case  the  inner 
edges  of  the  teeth  are  not  directed  toward  the  centre  of  the 
wheel. 

1st.  This  figure  represents  the  face  or  side  of  a  wheel  fifty- 
four  inches  in  extreme  diameter,  having  x  for  its  centre. 
After  describing  the  outer  circumference,  take  any  required 
distance  on  a  diameter,  as  3  d,  and  from  x,  with  the  radius 
x  d,  describe  the  circle  d  ef,  which  is  to  limit  the  length  of 
the  teeth. 

2d.  Divide  each  quarter  of  the  circumference  into  as  many 
equal  spaces  as  there  are  to  be  teeth  in  it.  From  the  centre, 
x,  describe  an  inner  circle,  a  b  c,  large  or  small,  according  to 
the  amount  of  bevel  that  is  to  be  given  to  the  teeth. 

3d.  From  the  points  of  division  in  the  circumference,  as 
#,#,4,  etc.,  draw  lines  tangent  to  (touching)  the  inner  circle, 
r.<5  seen  in  the  drawing — such  as  3  a,  8  c,  etc.  ;  and  also, 
from  the  same  points  of  division  in  the  circumference,  draw 
the  tooth  lines  3  g,  h  o,  etc.,  intersecting  the  former  tangen- 
tial lines  on  the  second  circle.  But  the  drawing  itself  will 
explain  every  thing  that  is  needed  to  be  understood. 

Fig.  58  represents  not  only  the  side  view  of  a  ratchet 
wheel,  but  the  dimensions,  also,  of  all  its  parts,  in  length, 
breadth,  and  thickness. 

Fig.  58  represents  a  ratchet  wheel  twelve  feet  in  extreme 
diameter,  having  a  rim  fifteen  inches  wide  and  six  inches 
thick,  into  which  thirty-two  teeth  are  cut,  to  a  depth  of  nine 
inches.  It  has  a  hub  forty-two  inches  in  diameter,  with  a 
central  cylindrical  opening  for  a  shaft,  or  axle,  twelve  inches 
in  diameter :  and  twelve  spokes,  of  three  by  six  inches,  and 
thirty-six  inches  long,  set  at  equal  distances  apart,  connect 
the  rim  and  the  hub. 

1st.  Here  x  is  the  centre  of  the  vertical  front  face  or  side 
of  the  wheel,  and  y  the  centre  of  the  farther  side.  As  the 
circle  1111,  described  from  x,  with  a  radius  of  six  feet, 
limits  the  points  of  the  teeth  on  their  front  face,  so  the  circle 
222%,  described  from  y,  with  the  same  radius,  limits  the 
points  of  the  teeth  on  the  farther  side  of  the  wTheel.  There- 
fore, if  we  divide  the  outer  circumference,  1111,  into  thirty- 


138  INDUSTRIAL   DRAWING.  [BOOK   NO.  III. 

two  equal  parts,  as  in  Fig.  57,  and  from  all  the  points  of  di- 
vision draw  diagonal  lines,  like  d  g,f%,  m  n,  etc.,  intersect- 
ing the  circle  2222,  we  shall  have  the  lines  of  the  edges 
of  all  the  teeth. 

2d.  We  get  the  required  bevel  for  the  front  edges  of  the 
teeth,  in  the  same  manner  as  in  Fig.  57,  by  describing  the 
circle  a  a  a  from  the  centre,  a?,  and  drawing  all  lines,  such  as 
d  CjfSj  m  p,  etc.,  tangent  to  that  circle.  But  these  latter 
lines  are  limited  by  the  circle  3  3  8,  which  we  describe  from 
x,  limiting  the  length  of  the  front  edges  of  the  teeth.  A 
circle,  4  4  49  described  from  ?/,  with  the  same  radius,  limits 
the  edges  of  the  teeth  on  the  farther  side ;  but  all  these 
edges,  like  g  A,  2  4,  n'r,  etc.,  whether  concealed  from  view 
or  not,  must  be  drawn  tangent  to  the  rear  circle,  b  b  b,  de- 
scribed from  ?/,  and  corresponding  to  the  front  circle,  a  a  a. 

By  following  these  simple  principles,  describing  all  the 
circles,  and  drawing  all  the  lines,  just  as  they  would  be  in 
the  real  wheel,  the  dimensions  of  every  tooth  are  accurately 
represented,  and  all  measure  precisely  the  same. 

3d.  The  twelve  spokes,  each  three  inches  wide  in  front, 
and  six  inches  in  thickness,  are  laid  out  by  dividing  each 
quarter  of  the  inner  circumference  of  the  rim  into  three 
equal  parts,  and  then  laying  out  the  width  equally  on  each 
side  of  these  divisions.  By  this  arrangement  of  the  spokes 
all  show  to  good  advantage. 

The  pupil  should  not  only  draw  this  wheel,  but  also  one 
of  a  different  size,  and  with  a  greater  number  of  teeth. 

Fig.  59  represents  a  chain-pulley  wheel,  the  links  of  the 
chain  being  adapted  to  fit  over  the  projecting  triangular 
cogs.  The  design  is  sufficiently  illustrated  by  the  drawing, 
without  the  necessity  of  any  description;  but  the  pupil 
should  fully  describe  the  wheel. 

PAGE  TWELVE.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

Fig.  60  represents  a  windlass,  viewed  diagonally  length- 
wise. It  consists  of  a  cylindrical  shaft,  XX,  twelve  inches 
in  diameter  and  eighty-eight  inches  in  length ;  four  inches 
from  the  two  ends  of  which  are  two  drums,  W  W,  each 
twenty-eight  inches  in  diameter  and  twelve  inches  in  thick- 


CABINET   PERSPECTIVE — CURVILINEAR   SOLIDS.  139 

ness.  Projecting  from  the  centre  of  the  circumference  of 
each  drum,  at  right  angles  to  the  axis,  are  eight  spokes  or 
levers,  three  inches  by  four  inches  in  size,  and  with  a  pro- 
jection of  ten  inches  beyond  the  drum. 

A  smaller  shaft,  or  axle,  Y  y,  six  inches  in  diameter,  pro- 
jects, centrally,  sixteen  inches  beyond  the  ends  of  the  larger 
shaft,  and,  immediately  beyond  the  ends  of  this  larger  shaft, 
rests  in  two  bracing  supports,  each  eight  inches  wide,  fitted 
to  receive  the  axle. 

At  A  the  front  drum  is  drawn  separately,  together  with 
the  end  of  the  shaft  and  the  projecting  axle,  but  without 
the  spokes  or  levers,  and  without  the  support  in  which  the 
axle  rests.  Here  a  is  the  centre  of  the  axle,  where  it  con- 
nects with  the  shaft,  and  also  the  centre  of  the  end  of  the 
shaft,  while  b  is  the  centre  of  the  front  face  of  the  drum,  and 
c  the  centre  of  the  farther  face  or  side. 

At  J?  is  a  separate  drawing  of  the  support  in  which  the 
front  axle  rests.  The  timber  forming  the  base  in  which  the 
braces  rest  is  forty  inches  in  length,  four  inches  in  vertical 
thickness,  and  eight  inches  in  width.  The  supporting  braces 
are  two  by  eight  inches,  and  about  twenty-three  inches  in 
length  above  the  base.  The  tie-brace  at  the  top  is  cut  out 
in  a  semicircular  form,  six  inches  in  diameter,  to  receive  the 
axle. 

At  C  is  a  representation  of  the  drum,  TFJ  with  the  spokes 
in  their  places.  It  is  here  drawn  separately  for  the  purpose 
of  showing  the  method  of  placing  the  spokes  accurately  in 
the  centre  of  the  cylindrical  surface  of  the  drum.  This  draw- 
ing may  be  made  in  the  following  order: 

1st.  Take  some  point,  a,  as  the  centre,  from  which,  with  a 
radius  of  fourteen  inches,  describe  the  circumference  of  the 
nearer  and  visible  end  of  the  drum ;  and  from  d,  twelve 
inches  diagonally  from  «,  on  the  axis,  with  the  same  radius, 
describe  the  farther  circumference,  only  half  of  which  is  vis- 
ible. 

2d.  Xow,  as  the  spokes  to  be  inserted  are  three  inches 
wide  on  their  face  or  front  side,  and  four  inches  diagonally, 
they  must  be  inserted  in  the  central  third  portion  of  the 
cylindrical  surface  of  the  drum.  We  therefore  take  -a  d, 


140  INDUSTRIAL   DRAWING.  [BOOK   NO.  III. 

representing  the  line  of  the  axis  of  the  drum,  and  divide  it 
into  three  equal  parts  by  the  points  b  and  c;  and  from  b 
and  c  as  centres,  with  a  radius  of  fourteen  inches,  we  de- 
scribe the  two  dotted  circles  encompassing  the  drum.  These 
two  circles  mark  out  the  central  third  part  of  the  cylindric- 
al surface  of  the  drum,  as  they  would  appear  to  the  eye  if 
the  drum  were  transparent.  The  spokes  must  therefore 
start  out  from  the  drum  between  these  circles,  and  touching 
them. 

3d.  These  spokes  we  may  consider  four  in  number,  equi- 
distant from  one  another,  passing  at  right  angles  through 
the  axis  of  the  drum,  and  presenting  eight  projecting  arms 
to  our  view.  For  convenience,  we  will  suppose  the  circum- 
ference of  the  nearer  dotted  circle  to  be  divided  into  eight 
equal  portions  by  the  vertical  line  1  2,  the  horizontal  line 
5  6,  and  the  two  diagonal  lines  8  4  and  7  8,  giving  us  the 
equidistant  points  /,  /i,  J,  m,  o,  q,  9,  and  10.  From  /,  as  a 
starting-point,  measure  off  any  desired  distance,  as/<7,  and 
take  g  as  the  point  for  the  intersection  of  a  corner  of  the 
spoke  with  the  dotted  line.  From  h  measure  off  the  same 
space  to  i,  and  the  same  from^'  to  A?,  from  m  to  ??,  from  o  to 
p,  etc.,  for  points  at  which  the  corners  of  spokes  pass  through 
the  circumference  of  the  drum. 

4th.  From  each  of  the  several  points,  g,  i,  7j,  ??,  p,  etc., 
measure  off  three  inches  backward  on  the  dotted  line,  as 
g  11,  i  12,  etc.,  for  the  width  of  the  face  of  each  spoke. 
Connect  the  points  that  are  directly  opposite,  as  Up,  g  13, 
etc.,  by  very  light  lines  extended  indefinitely,  every  pair  of 
which  will  pass  equidistant  on  each  side  of  the  central  point, 
b,  and  will  represent  the  front  faces  of  the  spokes  as  passing 
through  the  axis  of  the  drum. 

5th.  The  length  of  the  projecting  arms,  or  spokes,  may  be 
determined  by  circumscribing  a  circle  around  the  centre,  b. 
Here  the  circle,  which  may  be  called  the  nearer  face  circle 
of  the  spokes,  is  drawn  with  a  radius  of  twenty-four  inches, 
giving  ten  inches  for  the  length  of  the  spokes,  and  passing 
through  the  points  e,  8,  x,  etc.  Now,  as  the  spokes  are  four 
inches  in  diagonal  width,  we  describe  another  circle  with  the 
same  radius  from  c  (four  inches  diagonally  from  b),  passing 


CABINET   PERSPECTIVE — CURVILINEAR   SOLIDS.  Ill 

through  the  points  5,  3,  1,  8,  w,  y,  etc.,  thus  limiting  the  di- 
agonal width  of  the  spokes.  Suppose,  now,  that  the  corner 
face  lines  of  the  spokes,  as  g  e,  16  r,  etc.,  extend  to  the  nearer 
face  circle  e  8  x.  From  the  intersections  of  these  corner 
lines  with  this  nearer  face  circle,  giving  the  points  t,  r,  v,  z,  x, 
etc.,  we  draw  lines  diagonally,  as  t  u,  r  s,  e  I,  v  w,  x  y,  etc., 
which  give  us  the  side  boundary-lines  of  the  ends  of  the 
spokes. 

6th.  The  lines  16  17, 11  18, 12  19,  etc.,  showing  the  in- 
tersections of  the  sides  of  the  spokes  with  the  surface  of  the 
drum,  we  also  draw  diagonally— for  all  lines  that  are  par- 
allel in  the  real  wheel  must  be  parallel  in  the  drawing. . 

The  outlines  of  the  spokes  are  thus  completed— all  accu- 
rately drawn  in  their  proper  places,  as  projecting  from  the 
central  third  part  of  the  surface  of  the  drum. 

At  D  the  several  parts  described  in  the  sections  ^1,  B,  and 
C  are  put  together,  resting  on  a  timber  platform.  Let  the 
pupil  verify  the  following  measures  according  to  the  scale. 

1st.  The  distance,  x  y,  between  the  drums,  is  fifty-six 
inches.  The  distance,  c  d,  on  the  axis,  between  the  centres 
of  the  inner  faces  of  the  two  drums,  must  give  the  same 
measure. 

2d.  As  each  drum  is  twelve  inches  in  thickness,  the  dis- 
tance, r  z,  between  the  outer  faces  of  the  drums,  must  be 
eighty  inches.  The  distance,  bg,  between  the  points  on  the 
axis,  from  which  the  circumference  of  these  outer  faces  are 
described,  must  give  the  same  measure. 

3d.  As  the  large  shaft  extends  four  inches  beyond  the 
extreme  faces  of  the  two  drums,  the  length  of  the  shaft, 
whether  measured  diagonally  on  its  surface  or  on  its  axis, 
must  be  eighty-eight  inches.  The  farther  end  of  the  large 
shaft  is  concealed  from  view,  but  its  position  can  be  easily 
determined  by  counting  twenty-two  spaces  diagonally  from 
the  point  5. 

4th.  As  the  tops  of  the  bracing  supports  of  the  axle  are 
in  immediate  contact  with  the  ends  of  the  shaft,  the  dis- 
tance beticeen  the  inner  faces  of  the  tops  of  these  supports 
must  also  measure  eighty-eight  inches. 

5th.  As  each  of  the  bracing  supports  is  eight  inches  in 


142  INDUSTRIAL   DRAWING.  [BOOK    NO.   Ill, 

width,  the  distance,  s  t,  between  the  extreme  faces  of  the 
tops  of  these  supports,  must  be  one  hundred  and  four  inches. 
We  thus  get  the  point  t,  and  the  outlines  of  the  concealed 
top  of  the  farther  brace. 

Cth.  As  we  have  the  point  1  at  the  bottom  of  the  nearer 
brace,  we  now  know  that  the  corresponding  point  2  of  the 
farther  brace  must  be  one  hundred  and  four  inches  from  it 
diagonally — that  is,  twenty-six  diagonal  spaces.  The  dis- 
tance from  3  to  4  must  measure  the  same. 

We  have  been  thus  particular  in  describing  the  drawing 
of  this  figure  for  the  purpose  of  showing  that  every  part 
may  be  drawn  with  the  most  perfect  accuracy,  and  so  as  to 
show  its  exact  measurement.  With  the  aid  of  the  ruled 
drawing-paper  all  such  drawings  may  be  executed  with 
great  rapidity;  and  any  required  degree  of  modification, 
also,  from  the  plain  angular  forms,  may  be  given  to  them. 
Thus  the  spokes  may  be  round,  or  even  tapering,  and  light- 
er than  here  shown,  and  all  the  parts  may  be  modified, 
while  at  the  same  time  accuracy  of  representation  may  be 
adhered  to,  provided  the  very  plain  principles  of  this  mode 
of  representation  are  clearly  understood. 


DRAWING-BOOK    No.  IV. 


CABINET   PERSPECTIVE  —MISCELLANEOUS 
APPLICATIONS. 

I.  DIFFERENT  DIAGONAL  VIEWS   OF   OBJECTS. 

Ix  the  preceding  two  books  on  Cabinet  Perspective,  ob- 
jects have  been  represented  not  only  as  viewed  diagonally, 
with  their  principal  surfaces  in  vertical  and  horizontal  posi- 
tions, but  as  viewed  diagonally  from  one  particular  point, 
which  point  is  supposed  to  be  diagonally  at  the  right  of  the 
object,  and  above  it.  But  although,  for  the  purpose  of 
avoiding  confusion,  it  is  best  to  represent  all  objects  from 
this  one  point  of  view  whenever  it  can  be  done  to  advan- 
tage, yet  objects  may  be  represented  diagonally  in  cabinet 
perspective  equally  well  from  any  one  of  four  points  of 
view,  as  will  be  seen  from  the  drawing  on  the  first  page  of 
Book  No.  IV. 

PAGE  ONE. 

1st.  Upper  right-hand  view. — In  division  A  the  objects 
are  represented  as  viewed  diagonally  from  above,  and  at  the 
right,  the  same  as  in  all  the  drawings  in  the  preceding  two 
books.  The  arrow,  m,  is  represented  as  coming  from  the  po- 
sition, at  an  infinite  distance  away,  from  which  the  objects 
in  A  are  viewed. 

2d.  Lower  right-hand  view. — In  division  B  the  objects  are 
represented  as  viewed  from  below,  and  at  the  right ;  and 
here  the  arrow,  n,  is  represented  as  coming  from  the  posi- 
tion, at  an  infinite  distance  away,  from  which  the  objects 
are  viewed.  It  will  be  observed  that  the  bracket,/,  is  best 


144  INDUSTRIAL    DRAWING.  [iJOOK    NO.   IV. 

represented  in  its  true  position  from  the  point  of  view  here 
taken. 

3d.  Lower  left-hand  view.— In  division  C  the  objects  are 
represented  as  viewed  from  beloio,  and  at  the  left;  and  here 
the  arrow,  o,  is  represented  as  coming  from  the  point  from 
which  the  objects  are  viewed. 

4th.  Upper  left-hand  mew.  —  In  division  D  the  objects 
are  represented  as  viewed  from  above,  and  at  the  the  left / 
the  arrow,  p,  being  represented  as  coming  from  the  position, 
at  an  infinite  distance  away,  from  which  the  view  is  taken. 

Thus  two  of  the  views  are  represented  as  taken  from 
above,  one  from  the  right  and  the  other  from  the  left ;  and 
the  other  two  views  are  represented  as  taken  from  below,  in 
like  manner.  In  this  way  we  get  inside  views  of  the  curves 
of  the  four  brackets ;  but  all  of  the  brackets  might  have 
been  drawn  equally  well  according  to  the  plan  of  division  A, 
if  they  had  been  placed  in  proper  positions  for  the  purpose. 
In  A  the  bracket,  c,  is  inverted,  for  the  purpose  of  showing 
to  the  best  advantage  the  inside  curves ;  and  the  brackets 
/*,  y,  and  h  might  have  been  drawn  in  similar  positions. 


II.  GROUND-PLANS    AND    CABINET-PLANS    OF 
BUILDINGS. 

PAGE  TWO.— SCALE  OF  ONE  FOOT  TO  A  SPACE. 

Fig.  1  is  the  ground-plan  of  a  building,  whose  outer  walls, 
one  foot  in  thickness,  embrace  an  extent  of  eighteen  by  twen- 
ty-six feet.  It  will  be  seen  that  the  lined  paper  is  admira- 
bly adapted  to  ground-plans  or  surface  representations,  as 
it  is  only  necessary  to  adopt  any  convenient  scale,  when 
the  plan  may  be  laid  down  with  perfect  accuracy  without 
any  measurement,  by  simply  counting  the  spaces.  In  a 
ground-plan,  like  Fig.  1,  we  can  show  the  position  and  thick- 
ness of  the  walls  and  partitions,  and  the  width  and  position 
of  doors  and  windows,  etc. 

Fig.  2  is  a  cabinet-plan  of  the  walls,  doors,  windows,  etc., 
of  Fig.  1 ;  the  whole  being  represented  from  the  fotmda- 


CABINET   PERSPECTIVE — MISCELLANEOUS.  145 

tions  upward  to  the  extent  of  three  feet.  Observe  that  all 
the  measures  in  Fig.  2  are  the  same,  according  to  the  st-:ik' 
adopted,  and  the  principles  laid  down  for  cabinet  drawing, 
as  in  Fig.  1.  Thus,  in  Fig.  2,  the  length  of  the  side  wall,  1 2, 
is  twenty-six  feet ;  and  the  distance,  1  4,  is  eighteen  feet — 
the  same,  in  both  cases,  as  in  Fig.  1.  But  in  Fig.  2  it  is 
shown  that  the  windows  in  the  front  room  come  down  with- 
in six  inches  of  the  bottom  of  the  wall,  while  those  in  the 
back  room  come  down  to  within  two  feet  and  a  half  of  the 
bottom  of  the  wall.  These  are  features  which  can  not  be 
shown  in  a  ground-plan. 

Fig.  3  is  another  ground-plan,  representing  a  building 
with  two  rooms,  one  fourteen  by  sixteen  feetj  and  the  other 
thirteen  by  sixteen.  The  position  for  a  stair-way  is  given  in 
a  corner  of  the  large  room,  and  the  position  for  a  shelf  is 
given  in  the  other.  Rectangular  openings  for  fire-places 
are  also  given  in  the  partition  wall;  but  we  can  not  show 
any  thing  more  than  a  mere  ground -plan  in  this  kind  of 
drawing,  and  must  look  to  the  next  figure  for  any  thing  like 
details. 

Fig.  4  is  a  cabinet  view  of  the  plan  of  Fig.  3,  embracing 
not  only  the  ground-plan,  but  the  walls,  bench,  stair-way, 
platform  to  entrance,  etc. — a//,  in  fine,  that  belongs  to  the 
plan  of  the  building,  to  the  extent  of  three  feet  above  the 
stone  foundations,  while  the  height  of  the  stone  wall  above 
the  earth  is  shown  also. 

In  making  a  drawing  of  this  kind  it  is  best  to  begin  at 
the  top,  and  work  downward,  tracing  every  thing  lightly  at 
first.  The  walls  may  be  made  of  any  desired  height;  and 
the  only  objection  to  making  them  then-full  height  is  that 
the  front  wall  will  then  obscure  the  rear  wall.  It  is  well, 
sometimes,  to  draw  the  front  wall  and  the  right-hand  wall 
only  two  or  three  feet  high,  and  represent  the  other  two  of 
their  full  height,  with  beams,  joists,  etc.,  projecting  a  short 
distance  from  them.  Or  any  section  of  the  building  may 
be  represented  separately,  and  a  scale  larger  than  that  used 
for  the  walls  may  be  adopted  for  portions  that  require  a 
minute  representation.  In  this  manner  the  most  elaborate 
building  may  be  fully  represented,  even  to  the  most  minute 

G 


146  INDUSTRIAL    DRAWING.  [BOOK    NO.   IV. 

details,  and  with  far  greater  accuracy  than  could  be  attain- 
ed by  description  alone. 

Fig.  5.  At  A  are  represented  a  stair-way  and  platform, 
with  the  side  to  the  front ;  and  at  B  the  same  are  repre- 
sented with  the  stairs  presenting  the  front  view.  Observe 
that  the  measurements  in  the  one  case  are  the  same  as  in 
the  other — the  distance  5  2  being  the  same  as  6  .£,  an^  •*  $ 
the  same  as  3  7,  etc. 

PROBLEM   FOR   PRACTICE. 

(This  may  be  omitted  by  the  younger  pupils.} 

It  is  required  to  draw,  in  cabinet  perspective,  a  building  similar  to  the 
ground-plan  of  Fig.  3,  but  of  the  following  dimensions  :  Length  of  building 
(1  2\  forty  feet ;  width  (1  3\  twenty-four  feet ;  thickness  of  walls,  eighteen 
inches ;  room  on  the  left,  twenty  by  twenty-two  feet  in  the  clear ;  room  on 
the  right,  seventeen  by  twenty-two  feet  in  the  clear.  Draw  the  front  wall, 
division  wall,  and  wall  on  the  right-hand  side,  only  three  feet  high,  but  the 
wall  on  the  rear,  and  left-hand  side,  eleven  feet  high.  Put  in  the  openings 
for  the  windows,  a  and  b,  in  the  centres  of  the  ends  of  the  rooms,  three  feet 
from  the  floor,  four  feet  wide,  and  seven  feet  high ;  the  door,  c,  two  feet 
from  the  partition  wall,  four  feet  wide,  and  seven  feet  high  ;  let  a  beam  one 
foot  square  at  the  end  rest  upon  the  rear  wall  and  left-hand  side  wall ;  let 
a  beam  of  the  same  size  project  from  the  rear  beam  four  feet  over  the  par- 
tition wall ;  and  let  one  project  forward  from  the  upper  corner  above  2, 
four  feet;  and  let  one  extend  from  the  corner  above  <?,  four  feet  to  the 
right,  over  the  front  wall.  Carry  a  tier  of  four  shelves,  two  and  a  half  feet 
apart,  twelve  feet  long,  and  two  feet  wide,  across  the  window,  up  against 
the  rear  wall  and  right-hand  corner  of  the  room  on  the  right,  the  top  of  the 
lower  shelf  being  two  and  a  half  feet  from  the  floor.  Carry  a  flight  of  eight 
stairs,  four  feet  wide,  with  one-foot  risers  and  one-foot  tread,  up  against  the 
rear  wall  of  the  room  on  the  left,  onto  a  platform  four  feet  square  in  the  cor- 
ner ;  and  then  have  four  stairs  running  from  the  platform  toward  the  front, 
up  to  the  top  of  the  wall.  Let  there  be  one  supporting  post,  six  inches 
square,  at  the  angle  where  the  stairs  turn,  and  one  to  support  the  end  of  the 
last  stair,  three  feet  six  inches  from  the  wall,  and  coming  out  flush  with  the 
end  of  the  stair. 

Let  the  other  windows,  doors,  etc.,  be  similar,  in  size  and  position,  to  those 
in  Tig.  4. 

PAGE  THREE.— SCALE  OF  ONE  FOOT  TO  A  SPACE. 
Fig.  6  is  the  ground-plan  of  a  building  twelve  feet  in  ex- 
treme width,  and  thirty-four  feet  in  length,  with  a  platform 
entrance  to  the  front  of  two  by  six  feet,  and  three  steps, 


CABINET   PERSPECTIVE — MISCELLANEOUS.  147 

each  a  foot  wide,  ascending  to  it.  There  are  two  rooms, 
each  ten  by  twelve  feet,  with  a  hall  between  of  six  by  ten 
feet,  and  openings  four  feet  wide  in  the  partition  walls,  for 
slid  ing-doors,  leading  from  the  hall  into  the  two  rooms.  The 
jambs  of  the  fire-places  bevel  outward.  At  A  the  ground- 
plan  of  the  fire-places  is  drawn  on  a  larger  scale,  showing 
the  exact  bevel  of  the  jambs. 

Fig.  7  is  the  same  as  the  preceding  figure,  here  changed 
to  the  cabinet-plan,  and  drawn  to  the  same  scale  as  Fig.  6. 
The  ground-plan  of  the  walls  represented  in  Fig.  6  is,  in  Fig. 
7,  transferred  to  a  stone  foundation,  which  is  raised  one  foot 
above  the  ground  at  the  corner  .7,  and  two  feet  at  the  corners 
2  and  3.  Hence  the  front  steps,  which  have  six-inch  risers 
and  one-foot  tread,  must  rise  a  distance  of  two  feet. 

At  B  is  a  cabinet  view  of  the  fire-place,  drawn  to  the 
same  scale  as  the  ground-plan  in  A.  Observe  how  the  fig- 
ures of  the  one  correspond  to  the  figures  of  the  other  in  rel- 
ative position,  and  that  the  measures  are  the  same  in  both. 

Fig.  8  represents  a  series  of  platform  structures  of  different 
elevations.  Thus,  if  we  begin  with  the  level  at  J,  we  as- 
cend toward  the  left  an  inclined  or  sloping  plane  until  we 
rise  three  feet,  to  the  level  of  H.  Rising  six  inches  from  Jf, 
we  step  onto  the  platform  Gr  or  J5f/  four  steps,  each  of  six 
inches'  rise,  then  bring  us  onto  the  platform  I  or  F;  eight 
steps,  each  of  six  inches'  rise,  then  bring  us  to  the  level  of 
the  platform  Z>. 

Now  it  would  be  difficult  to  ascertain  whether  JEis  high- 
er than  D  or  not,  if  we  did  not  know,  ovfind  out,  the  height 
of  the  box-like  frame-work  on  which  the  flooring,  E,  rests, 
and  likewise  its  projection  beyond  the  sides  of  the  frame. 
The  size  of  this  frame-work  is  sixteen  feet  square ;  and  the 
floor,  here  represented  as  six  inches  in  thickness,  projects 
one  foot  on  all  sides,  except  toward  Z>,  where  there  is  no 
projection.  The  frame-work  (without  the  floor)  rises  four 
feet  above  the  corners  5  and  7,  and  six  feet  above  the  corner 
6.  Hence  the  upper  corners  of  the  frame-work  would  be 
seen  at  the  points  1  and  #,  the  former  four  feet  vertically 
above  5,  and  the  latter  six  feet  vertically  above  6;  and  the 
other  corners  at  the  points  3  and  Jh  at  the  intersections  of  the 


148  INDUSTRIAL   DBA  WING.  [BOOK   NO.  IV. 

dotted  lines.  Now  if  we  connect  these  four  points,  as  shoxvn 
in  the  drawing,  we  have  the  outlines  of  the  top  of  the  frame- 
work. 

We  have  now  to  put  a  floor  six  inches  thick  on  the  top 
of  this  frame-work,  and  projecting  one  foot  beyond  all  its 
sides  except  toward  D.  If  we  extend  4  1  one  half  a  diag- 
onal space,  that  is,  one  foot,  to  o,  the  latter  point  will  be  the 
lower  left-hand  corner  of  the  projecting  floor ;  and  the  point 
7i,  six  inches  above  it,  will  be  the  upper  left-hand  corner  of 
the  floor.  Extend  the  side  lines  of  the  top  of  the  frame  one 
foot  beyond  their  intersections  at  2  and  5,  and  we  shall  have 
points  through  which  the  lower  side  lines  of  the  projecting 
floor  wrill  pass,  to  make  the  other  lower  visible  corners  t  and 
v.  Draw  from  t  and  v  lines  six  inches  vertically  upward, 
and  we  shall  have  the  upper  right-hand  corners  of  this  pro- 
jecting floor.  In  order  to  see  fully  the  truth  of  this  de- 
scription, it  would  be  well,  first,  to  draw  the  outlines  of  the 
top  of  the  frame  lightly,  without  the  flooring,  E,  and  after- 
ward to  put  on  the  flooring. 

As  it  requires  eight  steps,  of  six  inches  each,  to  rise  from 
the  platform  I  to  Z>,  D  is  four  feet  above  I;  and  as  the 
frame-work  of  E\$  four  feet  high  above  the  level  of  the  cor- 
ner 5,  and  as  on  this  frame-work  is  placed  a  flooring  six 
inches  thick,  it  follows  that  D  is  six  inches  below  E.  The 
farther  end  of  D  is  even  with  the  farther  side  of  the  frame- 
work of  E. 

The  posts  a  and  b  are  flush,  on  both  their  outer  sides,  with 
the  corners  of  the  frame-work  above  which  they  are  placed  ; 
the  posts  d  and  r  are  flush  with  the  right-hand  side  of  the 
frame-work,  and  consequently  are  one  foot  from  the  right- 
hand  edge  of  the  flooring ;  while  the  posts  p  and  c  are 
flush  with  the  front  and  rear  sides  of  the  frame-work,  and 
are  one  foot  each  from  the  front  and  rear  edges  of  the 
flooring. 

The  posts  on  the  platform  E  are  three  feet  high  above 
the  floor.  The  posts  on  the  platform  D  are  also  three  feet 
high ;  but  on  the  latter  posts  is  a  rail  six  inches  thick,  so 
that  the  top  of  the  rail  is  even  with  the  top  of  the  railing 
ofM 


CABINET   PERSPECTIVE — MISCELLANEOUS.  149 

At  C  are  the  outline  walls  of  a  building,  the  outer  walls 
being  two  feet  in  thickness.  At  L  is  seen  the  opening  into 
a  pit,  extending  downward  below  the  flooring. 

Observe  that  the  window  openings  in  the  frame  of  E  are 
the  same  in  relative  position,  number,  size,  etc.,  on  the  right- 
hand  side  as  on  the  front,  and  that  there  is  the  same  real 
width  of  window-sill  represented  on  the  right-hand  side  as 
on  the  front. 

To  master  the  drawing  of  the  platforms  in  Fig.  8 — with 
the  projection  of  E  on  three  sides,  and  no  projection  to- 
ward D — with  the  arrangement  of  the  posts,  etc.,  the  whole 
should  be  drawn  on  a  larger  scale. 

PROBLEM   FOR   PRACTICE. 

Draw  a  figure  similar  to  Fig.  8,  but  with  the  following  measures :  Let 
the  frame  on  which  the  flooring  of  E  rests  be  twenty-seven  feet  square. 
Height  of  frame  from  corners  5  and  7  six  feet,  and  from  corner  6  eight  feet. 
On  this  frame  place  a  floor  one  foot  in  thickness,  and  one  foot  projec- 
tion on  all  sides  except  toward  Z>,  where  there  is  to  be  no  projection. 
Posts  of  platform  E  four  feet  high  above  floor,  six  by  twelve  inches  in  size, 
and  arranged  as  in  Fig.  8.  Platforms  I  and  F  six  feet  wide.  Platform  Z>, 
and  steps  ascending  to  it,  eight  feet  wide ;  and  a  sufficient  number  of  steps, 
each  six  inches'  rise  and  twelve  inches'  tread,  for  D  to  be  six  inches  below 
E.  Top  of  railing  of  D  to  be  even  with  top  of  railing  of  E.  Let  there  be 
four  window  openings,  or  recesses,  in  front,  and  the  same  number  on  side 
of  E,  each  three  feet  wide  and  four  feet  high,  a  foot  and  a  half  from  the 
top  of  the  frame ;  and  let  the  depth  of  the  recesses  be  one  foot.  These 
window  openings  to  have  three  feet  space  between  them,  and  those  nearest 
the  corners  to  be  three  feet  from  corners.  In  other  respects  make  the 
structure  similar  to  Fig.  8,  omitting  (7. 


III.  CYLINDRICAL    OBJECTS    IN  VERTICAL 
POSITIONS. 

The  representations  of  circles,  wheels,  cylinders,  etc.,  in 
cabinet  perspective,  have  thus  far  been  made  on  the  suppo- 
sition that  the  axes  of  the  cylinders,  etc.,  lie  in  a  horizontal 
position,  although  placed  diagonally  with  reference  to  the 
point  from  which  the  view  is  taken ;  and  the  vertical  cylin- 
drical ends  have  been  drawn  perfect  circles,  as  though  they 


150  INDUSTRIAL   DRAWING.  [BOOK   NO.  IY. 

were  directly  in  front  of  the  spectator.  This  is  the  case 
with  all  the  cylindrical  objects  represented  in  Book  No.  III. 
Thus,  referring  to  the  cabinet  cube  for  illustration,  we  have 
hitherto  represented  the  circle  as  drawn  on  the  front  ver- 
tical face  of  the  cube.  But  if  the  circle  were  to  be  drawn  on 
the  obliquely  viewed  top  or  side  of  the  cube,  the  circle  would 
have  the  form  of  a  particular  kind  of  ellipse*  as  the  follow- 
ing illustration  will  show. 


I.  ELLIPSES  ON  DIAGONAL  BASES. 

PAGE  POUR.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

[N.  B. — The  description  of  Fig.  9  should  be  read  over,  but  the  figure  need 
not  be  drawn.] 

Fig.  9.  We  have  here  drawn  a  cabinet  cube  of  twenty 
spaces  (forty  inches)  to  the  side,  the  nearer  vertical  face  of 
the  cube  fronting  the  spectator ;  and  on  this  front  face  of 
the  cube  we  have  described  a  circle  touching  the  centres  of 
the  four  sides  of  the  square.  As  this  circle  is  drawn  with 
a  radius  of  ten  spaces,  the  circumference  will  pass  through 
the  intersections  of  the  ruled  lines  of  the  paper  marked  2,  4-, 
8, 10, 14, 16,  and  20, 22,  in  addition  to  the  points  0, 6, 12,  and 
18,  making  twelve  points  in  the  intersections  of  the  ruled 
lines  through  which  it  will  pass.f  On  the  top,  and  also  on 
the  right-hand  side  of  the  cube,  we  have  also  represented  a 
circle  of  the  same  size  as  that  in  front — the  circumference 
i»a  each  case  touching  the  centres  of  the  four  sides  of  the 
aiagonal  square  within  which  it  is  drawn,  and  also  pass- 
'  ng  through  the  corresponding  points  2, 4,  8, 10,  IJf.,  16,  and 
20,  22.  The  representations  on  the  top  and  right-hand  side 
of  the  cube  are  ellipses  •  and  a  sufficient  number  of  points 
in  their  curves  may  be  known  to  enable  one  to  draw  the 
curves  with  great  accuracy. 

Thus,  take  the  construction  of  the  upper  ellipse  for  illus- 


*  An  ellipse  is  an  oval  or  oblong  figure,  which  corresponds  to  an  oblique 
view  of  a  circle. 

t  A  circle  drawn  with  a  radius  of  5,  10,  15,  20,  25,  or  30  spaces,  etc.. 
will  pass  through  twelve  points  in  the  intersections  of  the  ruled  lines.  This 
is  susceptible  of  geometrical  proof. 


CABINET   PERSPECTIVE MISCELLANEOUS.  151 

tration :  First,  rule  the  upper  surface  of  the  cube  to  corre- 
spond to  the  ruling  on  the  front  face.  Then  all  the  lines, 
and  their  intersections,  on  the  upper  surface,  will  correspond 
to  those  on  the  front  face.  Mark  any  required  number  of 
points  on  the  front  face,  through  which  the  circle  passes, 
and  mark  the  corresponding  points  on  the  upper  surface, 
and  then  through  these  latter  points  draw  the  ellipse,  and 
it  will  correspond  to  the  circle.  Thus,  if  the  ellipses  be 
accurately  drawn,  they  will  pass  through  the  points  2,  4, 
8, 10, 14, 16,  £0, 22,  etc.,  of  the  top  and  right-hand  side  of  the 
cube.  More  points  may  be  taken,  if  required,  and  thus  the 
ellipses  may  be  drawn  quite  accurately  by  hand.  This 
mode  must  give  an  accurate  cabinet  representation  of  a  cir- 
cle drawn  on  the  top  and  side  of  a  cube.  View  the  whole 
figure  through  the  opening  of  the  partly  closed  hand  for 
a  half- minute  or  so,  and  the  ellipses  will  gradually  appear 
as  perfect  circles. 

But  the  two  side  curves  of  each  of  these  ellipses  may  be 
quite  accurately  drawn  in  the  following  easy  manner.  Sup- 
pose we  wish  to  draw  the  ellipse  in  the  cabinet  square 
B  C  F  E.  Place  one  point  of  the  compasses  in  the  point 
B,  and  extend  the  other  to  the  point  a,  the  centre  of  the 
opposite  long  side ;  and  with  the  compasses  thus  extended 
strike  a  curve  across  B  F,  the  diagonal  of  the  square,  and 
dot  the  point  b.  Then,  with  one  point  of  the  compasses  in 
F)  and  with  the  same  stretch,  dot  the  point  d,  corresponding 
to  b.  With  the  compasses  still  spread  as  before,  first  with 
the  point  at  a;,  and  afterward  at  a,  strike  the  curves  inter- 
secting on  the  right  at  m.  Then,  with  one  point  of  the 
compasses  in  m,  strike  the  side  curve  a  a;  of  the  ellipse.  It 
may  be  prolonged  in  the  direction  of  a  to  7.  Then,  with 
the  points  of  the  compasses  respectively  in  18  and  12,  the 
centres  of  the  other  two  sides,  with  the  same  stretch  as  be- 
fore, strike  the  curves  that  intersect  at  n;  and  from  n  as  a 
centre  describe  the  other  side  curve  12  18,  prolonging  the 
curve  to  about  the  point  19.  Half  the  distance  from  d  to 
B  will  give  the  point  21  of  the  ellipse,  on  the  diagonal ; 
and  half  the  distance  from  b  to  J^will  give  the  correspond- 
ing point  9.  The  diagonal  distance  from  9  to  21  is  also 


152  INDUSTRIAL   DRAWING.  [BOOK   NO.  IY. 

equal  to  the  distance  from  B  to  a — the  distance  first  laid 
off  on  the  compasses. 

In  a  similar  manner  the  side  curves  of  the  upper  ellipse 
may  be  described,  by  laying  off  the  same  distance  as  be- 
fore, from  B  to  o,  and  describing  the  intersecting  curves 
from  x  and  o,  etc.  This  method  will  give  a  dose  approxi- 
mation to  the  true  side  curves. 

The  ei.d  curves  of  the  ellipse  should  be  drawn  by  the  eye, 
after  first  marking  the  points  of  the  curve  on  the  diagonal, 
as  before  directed.  Fig.  9  should  be  referred  to  as  a  guide 
for  drawing  the  forms  of  all  similar  end  curves  in  ellipses 
thus  situated  ;  but  yet  it  will  seldom  be  necessary,  and  not 
often  desirable,  to  draw  cabinet  ellipses  in  those  positions. 
It  will  generally  be  found  most  convenient  to  draw  the 
curves  either  in  the  positions  and  in  the  manner  shown  in 
Book  No.  III.,  or  after  the  following  plan. 


II.  ELLIPSES   ON    RECTANGULAR   BASES. 

Fig.  10.  Suppose  that  the  cube,  Fig.  9,  should  be  so 
viewed  that  neither  the  right-hand  side  nor  the  left-hand 
side  could  be  seen  at  all,  but  that  the  top  of  the  cube  should 
appear  directly  above  the  front  face,  and  of  the  same  width 
as  in  Fig.  9.  The  whole  would  then  be  seen  as  in  Fig.  10, 
in  which  the  front  face  is  the  same  in  all  respects  as  in  Fig. 
9 ;  and  the  top  has  the  same  width,  0  12,  as  o  A  in  Fig.  9 
— but  the  right-hand  side  of  the  cube  is  not  visible.  Now 
the  top  of  the  cube,  supposed  to  be  viewed  from  an  infinite 
distance,  is  rectangular;  and  it  is  of  the  same  length,  from 
left  to  right,  as  the  front  face ;  and  its  apparent  width,  0  12, 
is  just  half  the  width  or  height  of  the  front  face.  Hence 
the  top  or  upper  side,  A  B  C  D,  as  thus  viewed,  has  a 
length  twice  its  width ;  and  the  circle  that  should  be  drawn 
on  the  top  of  a  cube  thus  viewed  would  show  as  a  perfect  el- 
lipse, whose  lesser*  diameter,  0  12,  is  just  one  half  the  length 
of  the  greater*  diameter,  6  18. 

This  ellipse,  representing  a  circle  forming  the  end  of  a 

*  The  lesser  diameter  of  an  ellipse  is  called  its  conjugate  diameter,  and 
the  greater  is  called  its  transverse  diameter. 


CABINET    rEr.SPECTIYE [MISCELLANEOUS.  153 

vertical  cylinder,  may  be  drawn  after  the  manner  first  indi- 
cated in  Fig.  9,  by  dotting  points  to  correspond,  in  relative 
position,  to  the  points  marked  on  the  circle  below.  The  el- 
lipse is  here  drawn  in  this  manner,  by  dotting  the  points, 
and  then  drawing  the  curve  through  them  by  hand.  All 
the  points  in  the  ellipse  that  are  accurately  marked  in  this 
way  must  be  correct.  As  the  circle,  described  with  a  ra- 
dius often  spaces,  passes  through  the  twelve  points  0, 2,4,  £» 
8, 10, 12, 14, 16, 18, 20, 22,  so  must  the  ellipse  above  it,  if  accu- 
rately drawn,  pass  through  the  corresponding  twelve  points, 
numbered  in  like  manner. 

Side  Curves. — But  the  side  curves,  at  least,  of  this  and  of 
all  similar  ellipses,  may  be  very  accurately  drawn  by  the 
compasses  in  the  following  manner.  Place  one  point  of  the 
compasses  in  A,  and  extend  the  other  to  6,  the  middle  of 
right-hand  side  of  the  cabinet  square  A  B  C  D  (or  place 
in  .Z>  and  extend  to  18) ;  then,  still  continuing  the  point  in 
A,  with  the  other  point  strike  the  central  vertical  line,  12  0 
prolonged,  in  10.  Then,  with  one  point  of  the  compasses  in 
ic,  and  the  other  extended  to  12  above,  describe  the  curve 
9  12  15.  Then,  with  one  point  in  C,  and  the  other  extended 
to  18  (half  way  between  A  and  D),  strike  the  line  0  12  pro- 
longed above  12,  for  the  point  on  the  opposite  side  to  cor- 
respond to  the  point  w ;  from  which  point,  thus  found,  and 
with  the  other  point  of  the  compasses  extended  to  0,  de- 
scribe the  curve  3  0  21.  The  two  side  curves  will  thus  be 
drawn  very  accurately. 

End  Curves. — The  end  curves  may  be  best  drawn  by 
hand,  in  the  following  manner,  by  the  aid  of  guide  circles. 
Thus : 

On  the  line  6  18  take  any  point,  v,  so  that  v  6  shall  be 
equal  to  v  9,  and  from  v  describe  a  circle  passing  through 
the  points  9,6,3:  this  circle  will  then  serve  as  a  guide  for 
drawing  the  end  curve  of  the  ellipse,  which  must  pass  a  lit- 
tle within  the  circle,  and  at  the  same  time  be  a  natural  and 
graceful  continuation  of  the  side  curves.  The  other  end 
curve  is  drawn  in  a  similar  manner. 

Note. — The  representation  given  in  Fig.  10  may  be  called  upper  vertical 
rectangular  perspective.  But  if  the  spectator  were  horizontally  to  the  rifjht 

G2 


154  INDUSTRIAL   DRAWING.  [BOOK    XO.  IV. 

of  the  centre  of  the  front  circle,  at  the  proper  distance,  he  would  not  see  the 
top  of  the  cube,  but  the  right-hand  side  would  be  visible ;  and  on  that  side 
might  be  described  an  ellipse  like  the  one  now  seen  at  the  top  of  Fig.  1 0 ; 
only  the  longest  diameter  of  the  ellipse  would  then  be  in  a  vertical  position. 
This  might  be  called  right  horizontal  rectangular  perspective.  In  the  same 
manner,  if  the  spectator  were  horizontally  to  the  left  of  the  centre  of  the 
front  circle,  the  left  side  of  the  cube  might  be  seen,  and  on  that  side  might 
be  described  an  ellipse,  and  this  might  be  called  left  horizontal  rectangular 
perspective.  In  the  same  way  the  spectator  might  be  supposed  to  be  ver- 
tically below  the  centre  of  the  front  circle,  so  as  to  see  the  lower  side  of  the 
cube,  on  which  might  be  described  an  ellipse.  But  this  latter  would  be  a 
position  so  unusual  that  we  would  not  recommend  objects  to  be  thus  drawn. 
The  ellipse  is  drawn  in  precisely  the  same  manner  in  the  several  positions 
here  mentioned. 

The  position  of  the  horizontal  cylinder  in  Fig.  13  is  different,  as  regards 
the  spectator,  from  any  of  the  positions  above  described,  but  the  Eule  on  the 
next  page  applies  equally  to  all  of  them ;  and  even  this  does  not  differ  in 
principle  from  the  general  rule  (ELEMENTARY  RULE,  p.  85)  as  applicable 
to  all  drawings  in  cabinet  perspective. 

Fig.  11.  Iii  this  figure  we  have  drawn  the  ellipse  within 
the  cabinet  square  A  J3  C  _Z),  in  all  respects  like  the  ellipse 
of  Fig.  10.  The  point  y  is  the  point  from  which  we  de- 
scribe, with  the  compasses,  the  curve  3  w  21 ;  and  from  the 
point  z  we  describe  the  curve  9  12  15.  These  points  are 
found  in  the  same  manner  as  the  corresponding  points  for 
describing  the  ellipse  of  Fig.  10.  Suppose  this  ellipse  to 
be  the  upper  horizontal  end  of  a  vertical  solid  cylinder 
forty  inches  in  diameter.  The  distances  w  12  and  E  F 
therefore  alike  represent  forty  inches.  Suppose  the  cylin- 
der to  be  ten  inches  in  height,  and  that  we  wish  to  draw 
the  outline  of  the  visible  side  of  it  toward  the  spectator. 
The  cylinder  will  then  extend  downward  from  E,  w,  and  F^ 
five  spaces,  to  ./>,  r,  and  C.  ED  and  F  C  will  then  be  the 
vertical  side  lines  of  the  cylinder;  and  through  the  points 
J9,  r,  and  C  must  be  drawn  the  half  of  an  ellipse  correspond- 
ing in  all  respects  to  the  curve  E  w  F.  As  the  rectangle 
A  B  C  D  represents  the  square  inclosing  the  top  of  the 
cylinder,  so  the  rectangle  E  F  Gr  H  represents  the  square 
embracing  the  bottom  of  the  cylinder;  and  if  we  describe 
a  cabinet  ellipse  within  this  lower  cabinet  square,  it  will 
give  the  outlines  of  the  bottom  of  the  cylinder.  Therefore 


CABINET   PERSPECTIVE — MISCELLANEOUS.  155 

we  describe  an  ellipse  within  the  square  E  F  G  H,  in  the 
same  manner  that  we  described  the  ellipse  within  the  square 
ABCD.  The  visible  portion,  C  r  D,  of  this  ellipse,  whose 
side  curve  is  described  from  the  point  p,  we  have  drawn 
with  a  firm  line ;  the  other  portion,  Z>  t  C,  which  would 
be  invisible  unless  the  cylinder  were  transparent,  we  have 
drawn  with  a  slightly  dotted  line. 

Fig.  12.  Here  the  solid  cylinder,  the  method  of  drawing 
which  has  been  fully  described  in  Figs.  10  and  11,  is  shown 
as  completed  and  shaded.  Any  vertical  cylinder,  of  any 
given  dimensions,  may  be  easily  drawn  by  the  method  here 
described,  and  with  such  accuracy  that  all  its  parts  may  be 
readily  measured. 

Let  it  be  observed  that  as  the  rectangle  A  B  C  D  of 
Fig.  10  corresponds  to  the  rhombus  A  B  C  D  of  Fig.  9, 
therefore  the  line  C  B  of  Fig.  10  measures  the  same 
(in  cabinet  perspective)  as  the  diagonal  line  C  B  of 
Fig.  9;  and  0  12  and  D  A  of  Fig.  10  the  same  as  0  12 
and  D  A  of  Fig.  9,  etc.  Therefore  we  have  the  follow- 
ing rule  for  representing  objects  in  rectangular  cabinet  per- 
spective : 

RULE. — When,  in  drawing  objects  in  RECTANGULAR  cabinet 
perspective,  a  diagonal  space  is  changed  to  a  vertical  space, 
the  latter,  when  thus  used  to  represent  a  horizontal  distance, 
has  the  same  measure  as  the  former. 

Fig.  13.  The  upper  part  of  this  figure  is  drawn  on  the 
plan  of  Fig.  10  and  Fig.  12.  The  lower  part  is  a  horizontal 
cylinder  whose  end  directly  fronts  the  spectator.  Observe 
that  the  length  of  the  horizontal  cylinder,  as  measured  from 
1  to  2,  is  sixteen  inches ;  length  of  the  short  vertical  cylin- 
der only  four  inches. 

PROBLEMS   FOR   PRACTICE. 

1.  Draw  a  vertical  solid  cylinder  after  the  plan  of  Fig.  12,  whose  diame- 
ter shall  be  thirty-six  inches,  and  whose  -axis,  or  length,  shall  be  eighteen 
inches. 

2.  Draw  a  horizontal  cylinder  whose  diameter  shall  be  twenty  inches, 
and  length  twenty-four  inches,  with  one  end  directly  fronting  the  spectator, 
as  in  Fig.  13 ;  and  at  the  farther  end  of  this  cylinder,  and  starting  even 
with  the  top  of  it,  draw  a  vertical  cylinder  of  the  same  dimensions  as  the 


156  INDUSTRIAL   DRAWING.  [BOOK   NO.  IV. 

horizontal  cylinder.  The  horizontal  cylinder  is  to  project  forward  from  the 
front  vertical  side  of  a  cube  twenty  inches  square;  and  the  vertical  cylinder 
is  to  rise  from  the  upper  side  of  the  same  cube. 

3.  Draw  a  cube  twenty-four  inches  square,  in  right  horizontal  rectangular 
perspective.  Centrally  placed  on  the  right-hand  side  of  it,  extend  out 
horizontally  to  the  right  another  cube  sixteen  inches  square ;  and  on  the 
right-hand  side  of  .this  draw  a  horizontal  cylinder  twelve  inches  in  length 
and  sixteen  inches  in  diameter.  Shade  all  the  front  surfaces  light,  and 
the  right-hand  side  surfaces  dark.  Then  view  the  drawing  as  directed  on 
page  50,  except  that  the  eye  of  the  spectator  should  be  centrally  to  the 
right  of  it.  Also  place  the  length  vertically,  and  view  it  in  that  position 
until  it  seems  to  stand  out  boldly  from  the  paper. 

PAGE  FIVE.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

Fig.  14.  It  is  required  to  make  a  drawing  of  the  top  of 
a  hollow  vertical  cylinder  eighty  inches  in  extreme  diame- 
ter, and  whose  sides  are  eight  inches  in  thickness. 

1st.  Let  A  B  C  D  be  a  rectangular  cabinet  square  of 
eighty  inches  to  a  side.  Within  it  describe  an  ellipse 
touching  the  centres  of  its  four  sides,  after  the  manner 
shown  in  Figs.  10  and  11.  Here  w  will  be  the  point  for 
describing  the  side  curve  112;  and  x  for  describing  the 
other  side  curve  8  i  4;  while  t  and  s  are  the  points  for  de- 
scribing the  guiding  circles  for  the  outer  end  curves. 

2d.  Now,  as  the  walls  of  this  hollow  cylinder  are  to  be 
eight  inches  thick,  the  inner  circle  that  bounds  the  walls  on 
the  inside  must  be  eight  inches  within  the  outer  circle;  so  we 
take  o  r,  k  I,  v  p,  and  ij,  each  eight  inches  within  the  ellipse 
first  drawn,  and  complete  the  rectangular  cabinet  square 
E  F  G  II eight  inches,  on  all  sides,  within  the  outer  square. 
Within  this  smaller  square  describe  an  ellipse  touching  the 
centres  of  its  four  sides,  and  we  shall  have  the  outlines  of 
the  top  of  the  hollow  cylinder,  as  required. 

3d.  This  inner  ellipse  is  drawn  in  all  respects  like  the 
outer  one.  Thus  y  and  z,  found  as  before  shown,  are  the 
points  from  which  its  side  curves  are  described;  and  m  and 
n  are  the  points  from  which  the  guiding  circles  for  drawing 
the  ends  of  the  ellipse  are  described. 

4th.  Observe  that  the  walls  of  the  cylinder  appear  the 
thinnest  at  the  points  ij  and  k  I,  and  that  at  the  points  o  r 
and  v  p  they  appear  to  be  double  the  thickness  of  ij  and  k  /, 


CABINET   PERSPECTIVE — MISCELLANEOUS.  157 

as  they  would  naturally  appear  when  viewed  from  the  given 
point  above  and  in  front  of  the  cylinder. 

Fig.  15.  It  is  required  to  make  a  drawing  of  a  thin  flat 
ring,  forty  -  eight  inches  in  diameter  and  four  inches  in 
width. 

1st.  Take  the  horizontal  line  A  B,  of  twenty-four  spaces, 
representing  forty-eight  inches,  and  on  it  construct  the  rect- 
angular cabinet  square  A  JB  C  D.  A  D  or  JB  C  will  then 
represent  forty-eight  inches  also. 

2d.  Describe  the  ellipse  1234  within  this  square,  and 
touching  the  centres  of  its  four  sides.  This  ellipse  wrill 
represent  the  top  circle  of  the  ring.  The  central  line,  4  £, 
must  be  extended  upward,  so  as  to  get  the  point  from  which 
the  lower  side  curve  is  to  be  described. 

3d.  As  the  bottom  circle  of  the  ring  is  four  inches  below 
the  top  circle,  it  is  to  be  described  within  a  square  four 
inches  below  the  square  which  contains  the  top  circle. 
Therefore  lay  on0  a  second  square,  E  F  G-  J7,  four  inches 
below  the  top  square,  and  within  it  describe  the  ellipse 
5678,  touching  the  centres  of  its  four  sides  at  the  points 
5,  6,  7, 8.  The  outlines  of  the  drawing,  which  may  now  be 
shaded,  will  thus  be  completed. 

4th.  Notice  that  1  5  and  3  7  are  straight  vertical  lines, 
representing  the  width  of  the  ring,  four  inches;  and  that 
the  curve  9  6  10  would  touch  the  points  5  and  7  if  it  could 
be  seen  throughout. 

5th.  Observe  that  this  ring  is  drawn  as  having  no  appar- 
ent or  measurable  thickness. 

Fig.  16.  It  is  required  to  make  a  drawing  of  a  ring  forty- 
eight  inches  in  diameter,  four  inches  in  width  (or  height,  as 
here  viewed),  and  four  inches  in  thickness. 

1st.  Draw  the  cabinet  square,  A.  IB  C  D,  of  the  requisite 
dimensions,  and  within  it  describe  the  ellipse,  1 2  3  4>  for  the 
upper  outer  circumference  of  the  ring. 

2d.  Take  the  cabinet  square,  EFGr  H,  also  of  forty-eight 
inches  to  a  side,  four  inches  below  the  first-mentioned  square, 
and  within  it  draw  an  ellipse  in  all  respects  like  the  ellipse 
5  6  7  8  of  Fig.  15,  with  the  exception  that  only  the  front 
half  of  the  ellipse  in  Fig.  16,  as  denoted  by  the  figures  £,  8, 7, 


158  INDUSTRIAL   DRAWING.  [BOOK    NO.  IV. 

is  visible;  the  farther  half  being  concealed  by  the  thickness 
of  the  ring. 

3d.  In  order  to  get  the  lateral  thickness  of  the  ring,  take 
a  cabinet  square,  I J  I£  X,  four  inches  within  the  square 
A  BCD,  and  describe  the  ellipse,  9  10  11  1#,  touching  the 
centres  of  the  four  sides  of  this  inner  square.  The  thickness 
of  the  ring,  as  seen  on  its  upper  horizontal  edge,  will  then 
be  represented  by  the  space  between  the  two  ellipses  1234 
and  9  10  11  12. 

4th.  It  now  remains  to  describe  the  visible  portion  of  the 
lower  inner  circle  of  the  ring.  As  this  must  be  drawn  four 
inches  below  the  upper  inner  circle,  take  the  cabinet  square, 
U  V  W  JT,  four  inches  below  the  square  I  J  K  X,  and 
within  it  describe  an  ellipse  touching  the  centres  of  its 
sides.  Of  this  ellipse,  a  little  less  than  the  farther  half, 
op  r,  will  be  visible. 

Observe  the  quadrangle  (four  sided  figure)  at  each  corner 
of  the  drawing.  The  two  outer  corners  of  each  of  the  four 
quadrangles  are  the  corners  of  the  two  cabinet  squares 
within  which  the  two  outer  ellipses  are  drawn ;  and  the  two 
inner  corners  are  the  corners  of  the  two  squares  within 
which  the  two  inner  ellipses  are  drawn.  Similar  quadran- 
gles may  be  found  in  the  drawings  of  all  cylinders  on  a 
rectangular  basis ;  and  these  quadrangles  are  excellent 
guides  for  the  formation  of  the  squares  within  which  the 
several  ellipses  are  to  be  drawn.  Observe  that,  in  all  simi- 
lar drawings  of  hollow  cylinders,  the  distance  A  E,  or  B  F, 
or  C  6r,  or  D  H,  represents  the  height  of  the  cylinder. 

The  four  ellipses  described  in  this  figure,  representing 
the  four  circles  that  form  the  outlines  of  the  real  hollow 
cylinder,  are  all  the  guides  that  are  required  to  draw  any 
hollow  cylinder,  of  any  definite  .height,  diameter,  and  thick- 
ness of  walls,  in  the  vertical  position  represented  by  a  rect- 
angular cabinet  square,  as  distinguished  from  a  diagonal 
cabinet  square.  A  little  practice  will  enable  one  to  draw 
these  ellipses  with  great  facility,  while  the  ruling  of  the 
drawing-paper  insures  accuracy.  These  four  ellipses  an- 
swer to  the  four  circles  used  in  drawing  a  hollow  cylinder 
in  diagonal  cabinet  perspective. 


CABINET   PERSPECTIVE — MISCELLANEOUS.  159 

Fig.  17  is  the  same  in  outline  as  Fig.  16,  but  is  here  shaded, 
the  better  to  show  the  effect.  Observe  the  four  corner 
quadrangles,  which  are  here  retained  as  guides  for  the  four 
cabinet  squares  within  which  the  ellipses  are  to  be  drawn. 
The  points  from  which  the  upper  side  curves  are  drawn  are 
below  the  border  of  the  page.  Observe  that  the  point  1  is 
the  centre  of  the  upper  two  squares  and  ellipses;  and  that 
the  point  2  is  the  centre  of  the  lower  two  squares  and  el- 
lipses ;  while  the  line  1  2  represents  the  axis  of  the  short 
cylinder  or  ring. 

Fig.  18  represents  the  same  ring  as  in  Fig.  17,  but  with  a 
portion  of  the  front  of  the  ring  cut  out  vertically,  and  at 
right  angles  to  its  inner  and  outer  surfaces. 

PROBLEMS   FOR   PRACTICE. 

1.  Draw  a  vertical  hollow  cylinder  whose  extreme  diameter  shall  be 
fifty-two  inches ;  length  of  axis  (height  of  cylinder),  twenty  inches,  and 
thickness  of  walls,  six  inches. 

2.  Draw  two  hollow  cylinders  after  the  plan  of  Fig.  13,  page  4.     Let  each 
of  the  cylinders  be  thirty-two  inches  in  diameter,  sixteen  inches  in  length, 
and  the  walls  four  inches  in  thickness.     Remember  that  it  is  to  be  drawn 
on  a  scale  of  "two  inches  to  a  space."    This  measure  applies  to  both  ver- 
tical and  horizontal  lines  on  the  front  end  of  the  horizontal  cylinder,  while 
it  means  four  inches  to  a  space  on  lines  representing  horizontal  lines  run- 
ning directly  from  the  spectator.     Thus,  in  Fig.  13,  each  of  the  lines  1  3, 
and  4  $i  an(i  8  9  measures  twelve  inches ;  and  2  6  measures  four  inches ; 
but  G  7  measures  twelve  inches,  and  1  2  measures  sixteen  inches. 

PAGE  SIX.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

Fig.  19.  It  is  required  to  draw,  in  rectangular  cabinet  per- 
spective, a  hollow  vertical  cylinder,  forty  inches  in  extreme 
diameter,  fourteen  inches  in  height,  the  walls  four  inches 
thick,  and  to  be  encompassed  by  three  bands,  each  two 
inches  wide,  one  even  with  the  top,  one  even  with  the  bot- 
tom, and  the  third  intermediate  between  the  other  two. 

1st.  Draw  the  hollow  cylinder  after  the  manner  described 
in  preceding  figures.  Referring  to  Fig.  19,  let  the  pupil  ob- 
serve that  it  is  the  cabinet  square  A  J3  C  D  that  incloses 
the  top  of  the  cylinder,  and  the  square  E  F '  G  JET  that  in- 
closes the  bottom.  Hence  E  and  .F  are  fourteen  inches 
vertically  below  A  and  7?,  and  H and  G  are  fourteen  inches 


160  INDUSTRIAL   DBA  WING.  [BOOK    NO.  IV. 

vertically  below  D  and  C,  thus  measuring  two  inches  to  a 
vertical  space.  And  yet,  when  the  line  A  D  is  taken  as  the 
short  visible  side  of  a  rectangular  cabinet  square,  and  is  con- 
sidered as  running  horizontally  from  the  spectator,  being 
what  is  called  fore-shortened,  it  is  measured  as  four  inches 
to  a  space,  making  the  line  forty  inches  in  length.  This 
principle  must  be  kept  carefully  in  mind,  being  equivalent 
to  the  principle  of  the  measurement  of  "diagonal  spaces" 
in  Books  No.  II.  and  No.  III. 

2d.  To  describe  the  encircling  bands.  The  principle  is  the 
same  as  explained  in  describing  them  in  diagonal  cabinet 
perspective;  but  we  will  show  the  application  here  in  detail. 

As  w  is  the  point  from  which,  with  the  distance  w  4>  we 
describe  the  upper  and  outer  side  curve  embraced  in  1  4  8, 
so,  to  describe  the  side  curve  of  the  lower  side  of  the  first 
band,  we  move  the  point  of  the  compasses  one  space  (two 
inches)  downward,  from  w  to  r,  and  with  the  same  stretch 
of  the  compasses  as  before  describe  the  curve.  Then,  as 
the  next  band  is  four  inches  (two  spaces)  lower,  we  move 
the  point  of  the  compasses  two  spaces  (four  inches)  down- 
ward, from  r  to  s,  and  describe  the  upper  side  curve  of  the 
second  band.  So  continue  until  all  the  bands  are  drawn. 
The  end  portions  of  the  curves  are  drawn  without  the  com- 
passes, each  to  its  given  point  on  the  side  lines  1  5  and  3  7. 

We  have  also  described  six  curves  (portions  of  ellipses) 
on  the  visible  inner  side  of  the  cylinder,  between  m  and  99 
the  better  to  show  the  curvature  of  the  inner  side  of  the 
cylinder.  These  curves  are  two  inches  apart;  and  the  up- 
per and  the  lower  one  are  each  two  inches  from  the  top 
and  bottom  inner  curves  of  the  cylinder.  These  curves  are 
the  same  in  form  (so  far  as  seen)  as  the  upper  curve  I  m  n; 
and  as  the  side  portion  of  I  m  n  is  described  from  the  point 
z,  so,  to  describe  the  side  portions  of  these  six  curves,  carry 
the  point  of  the  compasses,  at  each  remove,  one  space  (two 
inches)  downward  below  z. 

Fig.  20  is  the  same  cylinder  as  represented  in  Fig.  19,  but 
is  here  drawn  in  diagonal  cabinet  perspective,  after  the 
manner  shown  in  Book  No.  III.  Observe  that  both  cylin- 
ders measure  the  same,  according  to  the  principles  of  meas- 


CABINET  PERSPECTIVE — MISCELLANEOUS.  161 

urement  applicable  to  each.  Observe,  also,  that  the  two 
cabinet  squares,  AB  CD,  E  F  G II,  of  Fig.  20,  correspond 
to  the  similarly  lettered  cabinet  squares  of  Fig.  19. 

Fig.  21.  It  is  required  to  draw  a  horizontal  wheel  of  the 
following  dimensions  in  rectangular  cabinet  perspective: 
Wheel  to  be  fifty-six  inches  in  extreme  diameter;  horizontal 
thickness  of  rim,  four  inches ;  width  of  rim  (or  height,  as 
it  lies  horizontally),  two  inches ;  a  hub  at  the  centre  six- 
teen inches  in  extreme  diameter,  same  vertical  thickness  as 
the  width  of  the  rim,  and  with  a  central  eight-inch  circular 
opening  through  it  for  the  axle.  There  are  to  be  eight 
spokes  between  the  hub  and  rim,  each  four  inches  in  hori- 
zontal thickness,  and  same  vertical  width  as  the  rim,  radia- 
ting from  the  centre  of  the  hub,  and  at  equal  distances  apart. 

1st.  Draw  the  upper  and  lower  ellipses  which  form  the 
boundaries  of  the  rim  in  the  manner  described  for  drawing 
hollow  cylinders  in  preceding  figures. 

2d.  Draw  the  boundaries  of  the  hub  in  the  same  manner. 

3d.  From  the  centre,  q,  with  a  radius  of  twenty-eight 
inches,  describe  a  circle  passing  through  the  extreme  points 
o  and  t  of  the  wheel.  Mark  the  points  w,  s,£>,  and  n  for  the 
diagonals  of  the  circle ;  then  these  points,  together  with  the 
centrally  dividing  points  m,  t,  r,  and  o,  will  divide  the  circle 
into  eight  equal  parts.  From  each  of  the  eight  given  points, 
beginning  at  m  and  moving  to  the  left  around  the  circle, 
lay  off  with  the  compasses  any  required  distance,  as  m  1 
and  n  2,  and  mark  accordingly  the  points  1,  2, 3,  4->  etc.,  for 
the  positions  in  the  wheel  in  which  the  spokes  are  to  be  rep- 
resented. Then,  as  the  spokes  are  to  be  each  four  inches 
in  horizontal  thickness,  take  with  the  compasses  a  width  of 
two  spaces,  and  lay  off  that  distance  on  the  circle  from  the 
points  /,  2, 3, 4, 5,  etc.,  and  we  shall  have  the  distances  1  x, 
2  y,  S  2, 4  -10,  etc.,  to  represent  the  horizontal  thickness  of  the 
spokes.  From  the  given  points  draw  the  dotted  lines  per- 
pendicularly toward  the  horizontally  dividing  line  o  t,  and 
from  the  points  where  any  pair  of  these  dotted  lines  strike 
the  outer  upper  ellipse  of  the  wheel, draw  lines  to  the  corre- 
sponding opposite  points  of  the  ellipse,  and  the  apparent  upper 
surface  width  of  the  spokes  will  be  given.  Observe  that  the 


162  INDUSTRIAL   DRAWING.  [BOOK   NO.  IV. 

nearer  these  spokes  approach,  in  lengthwise  position,  a  line 
running  directly  from  the  spectator,  the  wider  their  upper 
surfaces  appear,  and  the  narrower  their  side  views  appear. 

To  get  the  apparent  vertical  thickness  of  the  spokes,  and 
the  lines  of  their  junction  with  the  inside  of  the  rim  and  the 
outside  of  the  hub,  we  have  only  to  draw  lines  vertically 
downward  from  the  points  where  the  upper  lines  of  the 
spokes  intersect  the  ellipses  of  the  rim  and  hub,  and  then 
draw  the  lower  lines  of  the  spokes  as  shown  in  the  drawing. 
The  same  result  would  be  attained  by  supposing  the  spokes 
to  extend  through  the  rim,  as  shown  by  the  dotted  lines. 

Fig.  22  is  another  illustration  of  the  principles  of  dividing 
the  ellipse  into  any  number  of  equal  parts,  as  set  forth  in 
Fig.  21. 

It  is  required  to  make  a  drawing  of  a  vertical  tub,  of  the 
following  dimensions,  in  rectangular  cabinet  perspective: 
The  tub  is  to  be  twelve  inches  in  outside  height;  forty  inches 
in  extreme  diameter ;  having  a  bottom  one  inch  thick  from 
the  level  of  the  chimes  below ;  its  sides  to  be  vertical,  and 
composed  of  twenty-four  staves  of  uniform  width,  and  two 
inches  thick;  and  the  tub  is  to  be  encompassed  by  two 
hoops,  each  two  inches  wide,  the  upper  one  one  inch  from 
the  top  of  the  tub,  and  the  other  even  with  the  bottom. 

1st.  Draw  the  hollow  cylinder  for  the  walls  of  the  tub 
the  same  as  in  preceding  figures,  with  the  exception  of  the 
inside  lower  ellipse. 

2d.  Describe  the  semicircle,  6  0  6,  and  with  the  compasses 
divide  it  into  twelve  equal  parts.  This  is  easiest  done  by 
commencing  at  0,  and  dividing  each  quarter — as  from  0  to 
8,  then  from  3  to  6,  etc. — into  three  equal  parts.  This  will 
give  the  corresponding  points,  1, 2, 3,  4,  &9  6->  on  each  side  of 
0.  From  these  points  draw  dotted  lines  vertically  down- 
ward until  they  intersect  the  outer  upper  ellipse  ;  also  sup- 
pose these  dotted  lines  to  be  extended  until  they  intersect 
the  nearer  half  of  this  outer  upper  ellipse.  These  twenty- 
four  points  of  intersection  with  the  outer  ellipse  will  be  the 
points  of  division  for  the  apparent  outer  surface  width  of  the 
twenty-four  staves. 

3d.  The  top  dividing  lines  of  these  staves  must  all  be  di- 


CABINET   PERSPECTIVE MISCELLANEOUS.  163 

rected  toward  the  centre,  <?,  just  as  they  would  be  in  the 
real  tub,  and  the  intersections  of  these  lines  with  the  inner 
upper  ellipse  will  give  the  points  for  the  inside  vertical  di- 
visions of  the  staves.  It  will  be  noticed  that  the  staves  di- 
minish in  apparent  width  from  the  vertical  centre  to  the 
right  and  left  sides,  as  their  surfaces  are  seen  more  and 
more  obliquely,  but  that  they  increase  in  apparent  thickness, 
just  as  they  would  appear  to  do  in  viewing  the  real  tub. 

4th.  Observe  that  the  inside  height  of  the  tub,  as  meas- 
ured from  7  to  8,  is  only  eleven  inches,  as  required  to  be, 
while  the  outside  height  is  twelve  inches.  Therefore  the 
farther  lower  inner  ellipse  must  be  drawn  one  inch  higher 
than  it  would  be  drawn  if  the  tub  had  no  bottom  in  it. 

5th.  Observe,  also,  that  A  B  C  D  is  the  square  within 
which  the  upper  outer  ellipse  is  drawn,  and  that  E  F  Gr  H 
is  the  square  within  which  the  lower  outer  ellipse  is  drawn. 
The  distance  apart — from  A  to  E,  etc. — depends  upon  the 
height  of  the  cylinder.  In  Fig.  21  the  distance  is  two  inch- 
es, while  here  it  is  twelve  inches.  To  prevent  confusion  of 
lines,  the  lower  quadrangles  only  are  here  dotted. 

6th.  The  hoops  are  to  be  drawn  according  to  the  method 
explained  in  Fig.  19.  At  the  sides  they  should  project,  to 
the  extent  of  their  thickness,  beyond  the  vertical  side  lines 
of  the  tub. 

PROBLEMS   FOR   PRACTICE. 

1  Draw  a  wheel  similar  to  Fig.  21 ,  but  of  the  following  dimensions :  1"o 
be  sixty-eight  inches  in  extreme  diameter ;  horizontal  thickness  of  rim,  two 
inches ;  width,  or  vertical  height,  of  rim,  six  inches ;  a  hub  at  the  centre 
twenty  inches  in  extreme  diameter,  same  vertical  height  as  rim,  and  with  a 
sixteen-inch  central  opening  through  it  for  the  axle.  There  are  to  be  eight 
spokes  between  the  hub  and  rim,  each  two  inches  in  horizontal  thickness, 
and  same  width  vertically  as  the  rim,  radiating  from  the  centre  of  the  hub, 
and  at  equal  distances  apart. 

2.  Draw  a  tub  similar  to  Fig.  22,  but  of  the  following  dimensions :  To  be 
fifty-six  inches  in  extreme  diameter ;  sixteen  inches  in  outside  height ;  a 
bottom  that  comes  up  two  inches  from  the  level  of  the  chimes  below ;  its 
sides  to  be  vertical,  and  composed  of  thirty-two  staves  of  uniform  width, 
and  two  inches  thick ;  and  the  tub  is  to  be  encompassed  by  three  hoops, 
each  two  inches  wide — the  upper  one,  one  inch  below  the  tops  of  the  chimes; 
the  lower  one,  even  with  the  lower  ends  of  the  chimes ;  and  the  other,  four 
inches  above  the  lower  hoop. 


164  INDUSTRIAL   DEAWING.  [BOOK   NO.  IV. 

PAGE  SEVEN.— SCALE  OF  THREE  INCHES  TO  A  SPACE. 

Fig.  23.  The  same  crown-wJieel  that  is  shown  in  Fig.  56, 
page  10,  of  Book  No.  III.,  is  here  drawn  with  its  axis,  t  s, 
in  a  vertical  position.  The  rectangles  A  B  C  D  and 
E  F  G  II  represent  the  squares  A  B  C  D  and  E  F  G  II 
of  Fig.  56. 

Here  w  is  the  centre  of  the  tipper  horizontal  surface  of  the 
wheel,  across  the  ends  of  the  cogs ;  and  the  wheel,  as  in  Fig. 
56,  measures  eleven  feet  in  extreme  diameter  from  16  to  16. 
From  the  centre,  w,  we  therefore  describe  the  two  ellipses 
which  give  the  thickness  of  the  ends  of  the  cogs,  as  in  Fig. 
56  we  described  the  two  circles  from  the  corresponding  cen- 
tre, 10. 

Twelve  inches  below  w  we  take  the  point  y,  from  which, 
as  a  centre,  we  describe  the  two  ellipses  which  determine 
the  length  of  the  cogs,  and  also  the  thickness  of  the  wheel 
between  their  bases.  Six  inches  below  y  we  take  the  point 
u,  which  must  be  on  the  level  of  the  tipper  surfaces  of  the 
four  arms  which  support  the  rim ;  and  from  v  as  a  centre 
we  describe  the  ellipse  which  gives  the  limit  for  the  ends 
of  these  arms  at  their  upper  extremities.  As  these  arms  are 
to  be  twelve  inches  in  vertical  height,  or  thickness,  we  take 
a  point  a;,  twelve  inches  below  v,  from  which  we  describe 
the  ellipse  which  limits  the  lower  extremities  of  the  arms, 
and  also  gives  the  inside  lower  boundary  of  the  wheel. 

As  the  axle  on  which  the  wheel  turns  extends  fifteen 
inches  beyond  both  the  upper  and  the  lower  face  of  the 
wheel,  the  point  s  must  be  the  centre  of  one  end  of  the  axle, 
and  the  point  t  of  the  other;  and  as  the  diameter  of  the  axle 
is  twelve  inches,  we  have  the  dimensions  of  the  ellipse  de- 
scribed around  s  for  one  end  of  the  axle.  The  other  end,  t, 
of  the  axle  can  not  be  seen. 

The  point  v  is  the  centre  of  the  upper  end  of  the  hub, 
which  is  eighteen  inches  in  diameter. 

As  the  wheel  is  to  contain  thirty-two  cogs,  at  uniform 
distances  apart,  and  of  the  same  width  as  the  spaces  be- 
tween them,  we  divide  each  quarter  of  the  semicircle,  16  0  16, 
into  eight  equal  parts;  and  from  the  points  of  division  on 


CABINET   PERSPECTIVE MISCELLANEOUS.  365 

the  semicircle  draw  lines  vertically  downward  until  they  in- 
tersect the  outer  ellipse  16  17  16 ;  and  the  points  of  inter- 
section will  give  the  points  for  the  divisions  of  the  outer  el- 
lipse for  the  cogs  and  the  spaces  between  them.  From  these 
points  of  division,  lines,  as  d  w,  a  w,  etc.,  drawn  to  the  centre, 
w,  give  the  boundary-lines  for  the  sides  of  the  cogs  at  their 
ends.  In  a  similar  manner,  lines  drawn  like  b  y,  g  y,  p  y, 
etc.,  give  the  boundary -lines  for  the  sides  of  the  cogs  at 
their  bases. 

The  four  arms,  or  spokes,  which  support  the  rim  are  to  be 
six  inches  in  thickness,  and  twelve  inches  in  vertical  height, 
the  same  as  in  Fig.  56.  Here  we  have  placed  these  arms  in 
diagonal  positions,  because  they  thus  show  to  the  best  ad- 
vantage; and  the  only  remaining  point  now  is  to  give  the 
true  representation  for  their  thickness — six  inches. 

For  this  purpose,  from  v,  the  centre  of  the  ellipse  which 
passes  through  the  extremities  of  the  arms,  we  describe  a 
circle  (of  which  uj  is  a  part)  with  a  radius  of  sixty-three 
inches  (the  length  of  each  arm  from  the  centre,  v).  From 
the  point  i,  where  the  diagonal  v  i  intersects  the  circle,  draw 
the  vertical  line  i  r  to  intersect  the  before-mentioned  ellipse, 
and  it  will  intersect  it  at  r,  on  the  diagonal  of  the  ellipse. 
From  the  point  i  measure  off,  on  the  circle,  with  the  com- 
passes, *  h  and  ij9  each  equal  to  one  ruled  space,  and  there- 
fore each  representing  three  inches;  and  from  h  andj  draw 
lines  vertically  downward  to  intersect  the  ellipse.  The  space 
included  bet  ween  the  points  of  the  intersections  of  these  lines 
with  the  ellipse  will  then  represent  six  inches ;  and  if  lines 
be  drawn  from  the  points  of  intersection  parallel  to  r  v,  we 
shall  have  the  true  apparent  width  of  the  upper  surface  of 
this  one  arm  or  spoke  of  the  wheel.  The  true  width  of  the 
others  may  be  found  in  a  similar  manner;  or  it  may  be 
taken  by  the  compasses,  and  applied  to  the  other  arms. 

At  A  the  method  of  drawing  the  hub,  together  with  the 
arms  radiating  diagonally  from  it,  may  be  more  distinctly 
seen. 

Let  it  be  observed,  here,  with  what  perfect  accuracy  a  line 
drawn  vertically  downward  from  the  intersection  of  the  cir- 
cumference of  a  circle  with  its  diagonal  must  intersect  the 


166  INDUSTRIAL   DRAWING.  [BOOK    NO.  IV. 

ellipse  drawn  below  it  on  the  same  diameter  on  Us  diagonal. 
Owing  to  this  principle,  any  division,  and  any  measurement, 
made  on  the  circumference  of  the  circle,  according  to  what- 
ever scale  may  be  adopted,  may  be  accurately  represented 
on  the  ellipse. 

Thus,  in  B,  we  have  a  circle,  and  within  it  a  cabinet  el- 
lipse, both  drawn  on  the  same  diameter,  m  n.  Here  D  J?  is 
drawn  on  a  diagonal  of  the  circle,  and  II F  on  a  diagonal 
of  the  ellipse.  Observe,  also,  that  every  diagonal  of  the 
cabinet  ellipse  drawn  like  this,  in  a  rectangle,  is  wbat  we 
have  called  a  two-space  diagonal  line,  or  a  semi-diagonal. 
From  the  point  -?,  where  the  circumference  of  the  circle  in- 
tersects its  diagonal,  draw  a  line  vertically  downward,  and 
it  will  intersect  the  ellipse  on  its  diagonal  at  the  point  5. 
Now  the  measure  1  4  is  represented  on  the  ellipse  by  5  8, 
and  1  2  by  5  6 ;  and  if  1  4  is  equal  to  two  spaces,  or  six 
inches,  and  1  2  equal  to  the  same,  then  6  8  will  represent 
a  distance  of  twelve  inches ;  and  the  shaded  band,  6  t  s  8, 
which  extends  across  the  ellipse,  will  be  twelve  inches  wide. 
Remembering  that  the  cabinet  ellipse  represents  the  circle 
viewed  obliquely,  we  see  why  all  measures  on  the  circle 
must  have  their  accurate  representations  on  the  ellipse. 
Also,  we  see  from  the  three  representations,  Fig.  23,  and  A^ 
and  -Z?,  that  the  diagonal  of  a  cabinet  ellipse  must  be  drawn 
on  a  two-space  diagonal  line  that  passes  through  the  centre 
of  the  ellipse. 

PROBLEM    FOR    PRACTICE. 

Let  the  pupil,  af.ev  drawing  Fig.  23,  draw  another  crown-wheel  of  any 
given  dimensions  that  may  be  assigned. 

PAGE  EIGHT.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 
Fig.  24.  It  is  required  to  make  a  drawing,  in  rectangular 
cabinet  perspective,  of  a  tub  twenty-six  inches  in  vertical 
height,  forty-eight  inches'  extreme  measure  across  the  top, 
thirty-two  inches'  extreme  measure  across  the  bottom ;  the 
sides  formed  of  thirty-two  staves  of  equal  width,  and  two 
inches  in  thickness,  encompassed  by  two  hoops,  each  two 
inches  wide — one  of  them  two  inches  from  the  top  of  the 
tub,  and  the  other  one  inch  from  the  bottom ;  the  tub  to 


CABINET   PERSPECTIVE — MISCELLANEOUS.  167 

have  a  false  bottom  of  one-inch  stuff,  seven  inches  in  ver- 
tical height  from  the  level  of  the  top  of  the  tub ;  and  in  the 
centre  of  this  bottom  an  opening  twelve  inches  square. 

1st.  Mark  off  the  cabinet  square,  A  B  G  D,  forty-eight 
inches  to  a  side,  and  within  it,  and  touching  the  centres  of 
its  four  sides,  describe  an  ellipse  for  the  upper  outer  circum- 
ference of  the  tub. 

2d.  Two  inches  within  this  outer  ellipse  describe  another, 
and  we  shall  then  have  the  outlines  of  the  top  of  the  tub. 

3d.  From  m,  the  centre  of  the  top  diameter,  mark  off  m  n, 
twenty-six  inches,  for  the  vertical  height,  or  axis,  of  the  tub. 
The  point  n  will  then  be  the  centre  of  the  bottom. 

4th.  With  n  for  the  centre,  mark  off  the  square,  EF  G  H, 
of  thirty-two  inches  to  the  side,  and  within  this  square  de- 
scribe an  ellipse,  only  the  nearer  half  of  which,  123,  need 
be  firmly  drawn ;  and  this  part  of  the  ellipse  will  give  the 
visible  portion  of  the  outside  circumference  of  the  bottom 
of  the  tub. 

5th.  By  connecting  the  extreme  points  p  1  and  r  S  of  the 
diameters  of  the  upper  and  lower  ellipses,  the  lines  thus 
formed  will  give  the  side  outlines  of  the  tub  ;  and  if  these 
lines  be  continued  downward,  they  will  meet  in  a  point,  w, 
on  the  downward  continuation  of  the  axis  of  the  tub.  All 
the  outside  dividing  lines  of  the  staves  will  tend  toward  this 
point,  w,  while  the  inside  dividing  lines  will  tend  toward 
the  point  z,  above  w.  The  point  z  is  found  by  drawing 
lines  from  8  and  9  parallel  to  p  1  and  r  #,  until  they  meet 
at  a  point  in  the  downward  continuation  of  the  axis. 

6th.  Take  the  point  7,  seven  inches,  on  the  axis,  below  my 
and  7  will  be  the  centre  of  the  circle  (or  ellipse,  in  the  draw- 
ing) which  bounds  the  outer  upper  edge  of  the  false  bottom 
of  the  tub. 

If  the  sides  of  the  tub  were  vertical,  we  should  measure 
off,  from  the  inside  top  of  the  tub  at  the  farthest  point,  4  &> 
equal  to  m  7y  but  as  the  line  4  5  extends  downward  to- 
ward the  spectator,  and  is  in  reality  longer  than  m  7,  it 
must  appear  so  in  the  drawing.  The  point  5  (when  accu- 
rately found)  must  be  in  the  middle  point  of  the  farther 
side  of  the  cabinet  square  which  bounds  the  ellipse  that 


168  INDUSTRIAL   DRAWING.  [BOOK    NO.  IV. 

limits  the  false  bottom  of  the  tub,  and  which  ellipse  has  7 
for  its  centre.  To  obtain,  with  accuracy,  the  point  5,  and 
the  square  within  which  this  ellipse  is  to  be  drawn,  draw 
the  horizontal  line  from  7  toward  10,  and  it  will  intersect  a 
line  drawn  from  9  to  z  at  a  point  a.  The  distance,  7  a,  will 
then  be  one  half  of  the  longer  diameter  of  the  ellipse  which 
bounds  this  false  bottom  of  the  tub ;  and  if  7  a  be  twenty 
inches,  we  must  so  place  the  point  5  that  the  horizontal  dis- 
tance, 7  5,  shall  also  be  twenty  inches.  We  may  thus  eas- 
ily construct  the  square  within  which  the  ellipse  must  be 
drawn.  Centrally  within  the  ellipse  mark  out  the  opening, 
twelve  inches  square,  and  give  to  the  false  bottom,  on  its 
farther  side,  a  depth  or  thickness  of  one  inch. 

Vth.  The  divisions  showing  the  apparent  width  of  the 
staves  are  obtained  in  the  same  manner  as  in  Fio-.  23. 

O 

Fig.  25.  It  is  required  to  draw  an  octagonal  tub  (of  eight 
equal  sides),  having  an  extreme  upper  diameter  of  forty- 
eight  inches,  a  vertical  height  of  twelve  inches,  a  lower  ex- 
treme diameter  of  forty  inches,  the  sides  two  inches  in  thick- 
ness, and  the  whole  resting  on  a  two-inch  thick  octagonal 
bottom  that  projects  two  inches  beyond  the  sides. 

1st.  Mark  off  the  square,  A  B  C  D,  of  forty-eight  inches 
to  a  side ;  and  within  it,  and  touching  the  centres  of  its 
four  sides,  draw  lightly  an  ellipse,  of  which  ij  k  is  the  far- 
ther half.  As  the  central  vertical  line  j  n  (vertical  as  seen 
on  the  paper,  although  it  represents  a  horizontal  line),  the 
central  horizontal  line  i  /£,  and  the  two  diagonals  A  C  and 
B  Z>,  divide  the  circle  represented  by  this  ellipse  into  eight 
equal  parts,  so  the  intersections  of  these  lines  with  the  el- 
lipse give  us  the  eight  corners  of  an  octagon.  Connect  the 
points  or  corners  thus  found,  and  two  inches  within  this 
octagon  draw  another  in  like  manner,  and  we  shall  have  rep- 
resented the  thickness  of  the  walls  of  the  octagon.  This 
inner  octagon  may  be  drawn  without  first  drawing  its  el- 
lipse, simply  by  drawing  its  lines  parallel  to  the  surround- 
ing octagon,  and  two  inches  within  it. 

2d.  From  m,  the  centre  of  the  two  ellipses  and  of  the  oc- 
tagon just  described,  measure  twelve  inches  vertically  down- 
ward to  n.  Take  n  2  and  n  3,  each  twenty  inches,  and  on  2  3, 


CABINET   PERSPECTIVE. — MISCELLANEOUS.  169 

as  the  central  horizontal  line,  mark  off  the  cabinet  square 
E  F  G  H—E II or  F  G  being  half  the  length  of  E  F  or  of 
H  G.  Within  this  square  draw  an  ellipse  touching  the  cen- 
tres, 1,  3,  4,  2,  of  its  four  sides.  Divide  the  nearer  half  of 
this  ellipse  so  as  to  give  the  four  nearer  corners  of  an  octa- 
gon, in  the  same  manner  as  the  upper  octagon  M-as  formed. 
Connect  the  corners  of  this  lower  octagon  with  the  corners 
of  the  upper  octagon  by  the  lines  i  2,  n  4>  k  3,  etc. 

3d.  Two  inches  within  the  lower  square,  E  F  G  If,  inscribe 
an  ellipse,  in  the  same  manner  that  the  inner  upper  ellipse 
was  drawn,  and  find  the  corners  <5,  #,  7  in  the  same  manner 
that  the  corners  8,  9,  10  were  found.  Connect  these  lower 
corners  with  the  upper  corners.  This  inner  octagon  may 
be  drawn  by  simply  drawing  its  lines  parallel  to  the  outer 
octagon. 

4th.  Take  2  p  and  3  r,  each  one  space  (two  inches),  and 
from  p  and  r  draw  lines  parallel  to  the  border  octagonal 
lines,  and  we  shall  have  the  projection  of  the  bottom  two 
inches  beyond  the  walls  of  the  octagon,  thus  forming  an 
octagon  two  inches  beyond  the  other;  or  this  octagon  may 
be  formed  by  first  drawing  an  ellipse  on  p  r  as  a  central  base 
line.  The  two  inches'  thickness  of  the  bottom,  on  which  the 
octagonal  tub  rests,  is  easily  designated  by  simply  marking 
points  one  space  down  from  the  corners  above. 

Observe  that  the  thickness  of  the  walls  of  the  octagonal 
tub,  and  the  extension  of  the  platform  bottom,  naturally  ap- 
pear the  greatest  on  the  extreme  right  and  left  sides  of  the 
drawing.  Observe,  also,  that  the  farther  inner  side  of  the 
tub,  as  measured  by  the  line  9  6,  appears  deeper  than  the 
nearer  side  as  measured  by  the  line  n  4.  The  cause  of  this 
is  that  the  line  9  6  is  more  nearly  at  right  angles  to  the 
line  of  vision  than  the  line  n  4- 


III.  ARCHES  IN  DIAGONAL  PERSPECTIVE. 

Figs.  26  and  27  are  drawn  in  diagonal  cabinet  perspec- 
ve ;  but  it  is  the  upper  left-hand  view  of  them  that  is  here 
given,  as  explained  on  page  144,  and  illustrated  at  J9,  page  1, 
of  Book  No.  IV. 

H 


170  INDUSTRIAL   DRAWING.  [BOOK   NO.  IV. 

Fig.  26.  The  lightly  shaded  portion  of  Fig.  26  represents 
the  face  of  a  pointed  archway  fronting  the  spectator,  and  is 
therefore  drawn  in  its  true  relative  proportions:  the  base 
line  of  the  archway,  a  d,  being  thirty-two  inches,  and  the 
height,  e/,  thirty-eight  inches. 

The  outer  curve  line,  af9  of  the  arch,  is  here  drawn  by  the 
compasses,  from  a  point  twenty-six  spaces — fifty-two  inches 
— to  the  left  of  a;  or  it  may  be  drawn  from  any  other  point 
that  will  give  a  graceful  curve  connecting  a  and  f*  but 
having  drawn  this  curve,  the  opposite  curve,  d  f9  must  be 
made  to  correspond  to  it.  Then  the  inner  curves,  p  h  and 
g  h,  must  be  drawn  from  the  same  points  from  which  the  two 
outer  curves  were  drawn,  but  each  with  a  radius  of  twenty- 
four  spaces.  The  curve  v  h  must  be  drawn  from  a  point 
twenty-four  spaces  to  the  left  of  the  corner  v. 

As  Fig.  26  is  a  front  view  of  the  arch,  it  is  very  easily 
drawn. 

Fig.  27  is  a  diagonal  view  of  the  front  of  Fig.  26.  Here 
the  diagonal  figure,  a  b  c  d,  must  measure  the  same  as  the 
corresponding  rectangle  of  the  preceding  figure ;  and  the 
curve  a  f  must  be  drawn  by  hand,  and  be  made  to  pass 
through  points  corresponding  to  those  through  which  the 
curve  a  f  of  Fig.  26  passes.  Thus  the  points  n  and  m  of 
Fig.  26  are  respectively  four  and  eight  inches  from  certain 
points,  10  and  1^  in  the  corner  line  ab  ;  and  the  correspond- 
ing points  n  and  m  of  Fig.  27  must  be  similarly  situated. 
So  any  point  in  any  one  of  the  curves  of  Fig.  26  should, 
if  accurately  drawn,  have  its  corresponding  point,  by  meas- 
ure, in  a  corresponding  curve  of  Fig.  27.  This  principle 
should  apply  accurately  throughout  the  two  figures — the 
measures  from  point  to  point  in  the  one  being  the  same 
(according  to  the  principles  of  measurement)  as  the  meas- 
ures between  corresponding  points  in  the  other.  Thus  the 
two  points  n,  n,  in  Fig.  26  are  each  twelve  inches  horizontally 
distant  from  the  point  w ;  the  corresponding  points  n,  n  in 
Fig.  27  are  also  each  twelve  inches  (three  diagonal  spaces) 
distant  from  their  intermediate  point  10.  The  under  side 
of  the  arch  must  measure  four  inches,  horizontally,  in  any 
part  of  it.  So  the  outside  of  the  arch,  from  a  r  tofj,  must 
measure,  in  any  part,  four  inches  horizontally. 


CABINET   PERSPECTIVE — MISCELLANEOUS.  171 

Having  Fig.  26  as  the  pattern,  and  marking  any  required 
number  of  points  in  its  curves,  it  is  easy  to  mark  corre- 
sponding points  in  Fig.  27,  and  draw  the  curves  through 
them. 

Fig.  28,  drawn  in  diagonal  perspective,  consists  of  a  rect- 
angular box,  open  at  top  and  bottom,  thirty-two  inches  in 
outside  width,  sixty-four  inches  in  length,  thirty-four  inches 
in  height,  and  walls  four  inches  thick,  having  an  archway 
opening  through  both  ends,  and  two  archways  on  a  side. 
The  side  archways  are  the  same  in  size  as  those  at  the  ends; 
and  the  face  of  the  archway  in  the  front  end,  being  drawn 
in  its  true  relative  proportions,  is  the  guiding  pattern  for 
drawing  the  others.  Let  the  pupil  describe  the  dimensions 
of  the  arch,  method  of  drawing  the  side  arches,  etc.  If  the 
pupil  will  follow  the  directions  given  for  Fig.  27,  he  will 
probably  draw  Fig.  28  with  more  accuracy  than  it  is  given 
in  the  book. 

The  principles  here  illustrated  will  be  a  sufficient  guide 
for  the  drawing  of  all  arches  in  diagonal  perspective.  First 
draw  an  arch  of  the  required  dimensions  in  its  true  relative 
proportions,  with  its  face  fronting  the  spectator:  it  will 
then  be  easy  to  make  a  diagonal  view  of  the  same  arch — 
transferring  the  measures  of  the  one  to  the  other. 


IY.  SEMI-DIAGONAL    CABINET    PERSPECTIVE. 
PAGE  NINE.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

In  what  has  been  called  Diagonal  Cabinet  Perspective,  a 
cubical  block  is  supposed  to  be  viewed  in  such  a  manner 
that  the  horizontal  corner  lines  of  the  top  and  bottom  of  the 
cube  that  run  from  the  spectator  seem  to  rise  diagonally 
at  r.n  angle  of  forty-five  degrees — that  is,  half  way  from  the 
horizontal  to  the  vertical.  Thus  in  Fig.  29,  if  the  cube  were 
drawn  in  diagonal  cabinet  perspective,  its  upper  horizontal 
face  would  be  bounded  by  the  lines  1  £,  2  3t  3  7,  and  7  1; 
and  5  4  would  be  its  visible  lower  side  line. 

Now  suppose  that  the  eye  of  the  spectator  should  be 
lowered  so  that  the  line  2  3  should  be  brought  down  and 


172  INDUSTRIAL   DRAWING.  [BOOK   NO.  IV. 

made  to  coincide  with  the  line  89:  then  1  2  would  be 
brought  down  to  1  8,  and  7  8  to  7  9,  and  5  4  to  5  10;  and 
thus  the  lines  1  8,  7  9,  and  5  10  would  be  in  the  direction  of 
semi-diagonals,  or  two-space  diagonals,  as  shown  in  Book 
No.  I.,  page  2.  The  front  face  of  the  cube  would  remain 
the  same  as  in  diagonal  perspective ;  but  the  top  and.  side 
would  be  changed  as  shown  in  Fig.  29.  This  is  what  is 
called  "  $e/m-Diagonal  Cabinet  Perspective;"  and  it  is  spe- 
cially adapted  to  the  drawing  of  buildings  or  other  large 
structures  in  full  elevation,  as  hereafter  shown. 

In  this  modification  of  Cabinet  Perspective,  the  principles 
of  measurement  remain  the  same  as  in  the  general  system 
before  explained.  Thus,  in  diagonal  perspective,  the  line 
1  2,  according  to  the  scale  for  the  page,  measures  sixteen 
inches:  but  the  line  18  must  also  measure  sixteen  inches :  the 
line  1 2  extends  from  the  point  1,  diagonally,  to  the  fourth  ver- 
tical line  at  2;  and  the  line  1  8  extends  from  the  point  1, 
semi-diagonally,  to  the  fourth  vertical  line  at  8 — that  is,  the 
line  1  2  passes  over  four  spaces,  and  the  line  1  8  also  passes 
over  four  spaces ;  and  in  each  case  a  space — whether  a  diag- 
onal or  a  semi-diagonal — measures  four  inches.  If,  therefore, 
we  consider  that  each  sem /-diagonal  (that  is,  the  distance  on 
the  line  1  8  from  one  vertical  line  to  another)  measures  the 
same  as  a  diagonal,  the  rule  of  measurement  is  unchanged. 
The  semi-diagonal  cube,  Fig.  29,  therefore  measures  sixteen 
inches  to  a  side,  the  same  as  the  dotted  diagonal  cube. 

Fig.  30  represents  a  cubical  block,  of  twenty-four  inches 
to  a  side,  from  each  upper  corner  of  which  is  taken  away  a 
cubical  block  eight  inches  square.  There  is  an  eight-inch 
square  opening  cut  through  the  block,  centrally,  from  the 
lower  part  of  the  front  side ;  and  also  an  opening  of  the 
same  dimensions  on  the  right-hand  side. 

Fig.  31  is  a  frame -work  in  semi -diagonal  perspective. 
The  central  post  measures  eight  inches  square  at  the  ends; 
and  each  half  of  it  is  thirty-six  inches  in  length  on  each 
side  of  the  cross-beam.  The  cross-beam  is  four  by  eight 
inches,  and  sixty -four  inches  in  length,  while  the  braces 
would  measure  four  inches  square  at  the  ends  if  the  ends 
were  squared.  Inasmuch  as  the  side  of  the  frame  fronting 


CABINET   PERSPECTIVE — MISCELLANEOUS.  173 

the  spectator  is  drawn  in  its  true  relative  proportions,  the 
same  as  all  front  views  in  the  cabinet  perspective  of  plane 
solids,  therefore  all  lines,  drawn  in  any  direction  whatever, 
on  any  portion  of  this  front  view,  are  measurable  by  the 
scale  of  two  inches  to  a  space.  Hence  the  line  a  £>,  drawn  at 
right  angles  across  the  brace,  must  be  the  true  measure  of 
the  width  of  the  brace ;  and,  according  to  the  scale,  it  is 
found  to  measure  two  spaces,  or  four  inches.  The  line  c  cl, 
beino-  one  semi-diasronal,  measures  four  inches.  We  thus 

O  O  7 

obtain  the  size  of  the  timber  used  for  the  brace. 

The  extreme  length  of  a  brace,  1  3,  or  its  equal  in  length, 
4  £,  may  be  found  by  actual  measurement ;  for,  as  the  line 
1  3  is  in  a  plane  at  right  angles  to  the  line  of  vision,  it  may 
be  measured  by  the  scale.  Thus  measured,  it  will  be  found 
that  the  line  1  S  is  equal  to  twenty  spaces — equal  to  forty 
inches.  Or  it  may  be  measured  by  the  rule  for  the  extrac- 
tion of  the  square  root.  Thus  1  2  3  is  a  right-angled  tri- 
angle; and  therefore,  if  we  add  together  the  squares  of 
the  sides  1  2  and  1  3,  and  extract  the  square  root  of  their 
sum,  we  shall  get  forty  inches  for  the  length  of  the  line  1  3. 
Try  it. 

Observe  that  the  braces  are  flush  with  the  farther  side  of 
the  central  post,  and  with  the  farther  side  of  the  cross-beam. 
The  dotted  continuations  of  the  lines  of  the  braces  show  the 
attachment  of  the  braces  to  those  sides  of  the  post  and  cross- 
beam which  are  invisible  to  the  eye. 

Fig.  32  is  the  same  as  Fig.  31,  but  so  placed  that  the  front 
view  of  Fig.  31  is  made  the  semi-diagonal  view  in  Fig.  32. 
Let  the  pupil  test  all  the  measurements,  and  see  that  they 
are  the  same  in  both  figures.  Thus  the  line  6  7  in  Fig.  32 
measures  the  same,  according  to  the  principles  of  measure- 
ment adopted,  as  the  line  6  7  in  Fig.  31.  Fig.  32,  however, 
does  not  give  so  good  a  view  of  the  frame  as  is  shown  in 
Fig.  31. 

Fig.  33  is  the  same  as  Fig.  95  of  Book  No.  II.,  but  is  here 
changed  from  diagonal  to  se??n'-diagonal  perspective.  Ob- 
serve that,  if  the  same  scale  be  adopted,  the  measurements 
will  be  the  same  in  both  cases. 

Fig.  34  is  a  hollow  cylinder  sixteen  inches  in  length  and 


174  INDUSTRIAL   DRAWING.  [BOOK   NO.  IV. 

eight  inches  in  extreme  diameter,  having  its  walls  two 
inches  in  thickness.  The  length  of  the  cylinder  may  either 
be  measured  on  its  axis,  a  b,  or  any  where  semi-diagonally 
on  its  circumference — as  on  the  line  c  d. 

Fig.  35  is  a  cylinder  eight  inches  in  length  and  sixteen 
inches  in  diameter,  having  an  opening  eight  inches  square 
through  it  centrally  lengthwise. 

PROBLEMS   FOR   PRACTICE. 

1.  Draw,  in  semi-diagonal  perspective,  a  cube  thirty-two  inches  square, 
and  take  from  the  centres  of  each  of  its  three  visible  sides  a  piece  twenty- 
four  inches  square  on  the  face,  and  four  inches  in  thickness. 

2.  Draw  a  frame  the  same  as  Fig.  31,  with  the  exception  that  the  braces 
are  to  be  placed  flush  with  the  front  side  of  the  central  post  and  cross-beam. 

PAGE  TEX.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

Fig.  36  represents  a  cube  having  a  circle  described  on  its 
top,  and  also  one  on  each  of  its  two  visible  sides,  the  whole 
being  drawn  in  semi-diagonal  perspective.  The  representa- 
tive ellipse  on  the  right-hand  side  of  the  cube  is  of  the  same 
width  as  in  diagonal  perspective,  but  not  of  the  same  pro- 
portions; but  the  ellipse  on  the  top  has  only  half  the  width 
that  it  would  have  in  diagonal  perspective.  The  front  side 
being  a  square,  the  circle  is  there  drawn  as  a  circle,  in  its 
true  proportions.  See  Fig.  29,  and  explanation,  for  the  rel- 
ative proportions  of  the  three  visible  sides  of  the  cube. 

As  the  upper  side  of  the  cube  is  divided  into  the  same 
'•  inmber  of  semi-diagonal  squares  that  the  front  side  contains 
'cgular  squares,  the  former  representing  the  latter,  therefore 
the  ellipse  drawn  on  the  top  must  pass  through  the  same 
corresponding  points  in  the  semi-diagonal  squares  that  the 
circle  passes  through  in  the  squares  of  the  front  side,  on  the 
principle  that  was  explained  in  the  drawing  of  Fig.  9.  The 
ellipse  on  the  top  would  be  best  drawn  by  first  dotting  the 
principal  points  through  which  it  must  pass, and  then  drawing 
the  curve  carefully  by  hand  with  a  sharp-pointed  hard  pencil. 

The  ellipse  on  the  side  might  be  drawn  by  the  same  meth- 
od, on  the  principle  explained  with  reference  to  Fig.  9 ;  but 
the  side  curves  may  be  drawn  with  great  accuracy  by  the 
compasses  in  the  following  manner: 


CABINET   PERSPECTIVE MISCELLANEOUS.  175 

1st.  From  the  point  1,  the  centre  of  the  side  C  f]  with  the 
radius  1  B,  describe  the  indefinite  curve  B  5.  2d.  From 
the  point  B,  with  the  radius  B  2,  describe  the  curve  2  3,  in- 
tersecting the  curve  B  5  at  the  point  3.  3d.  From  the  point 
3,  with  the  radius  3  2,  describe  the  curve  2  4.  4th.  From 
the  point  4,  with  the  radius  4  !>>  describe  the  side  curve 
617,  which  may  be  extended  to  within  two  inches  of  the 
line  C  B  with  accuracy. 

5th.  From  the  point  2,  the  centre  of  the  side  B  E,  with 
the  radius  2  F,  describe  the  indefinite  curve  Fp.  6th.  From 
the  point  JF\  with  the  radius  F 1,  describe  the  curve  1  n. 
7th.  From  the  point  n,  with  the  radius  n  -/,  describe  the 
curve  1  m.  8th.  From  the  point  m,  with  the  radius  m  2, 
describe  the  side  curve  9  2  10 ;  which  may  be  extended  to 
s bout  the  point  11  with  accuracy. 

Thus  the  two  side  curves  of  the  ellipse  will  be  drawn,  to- 
gether with  portions  of  the  two  end  curves.  The  remaining 
portions  of  the  end  curves  must  be  drawn  by  hand  to  the 
points  v  and  «?,  taking  care  that  they  pass  through  points 
in  the  small  semi-diagonal  squares  corresponding  to  the 
points  through  which  the  circle  on  the  front  passes.  Let 
this  figure  be  viewed  intently  through  the  opening  formed 
by  the  partially  closed  hand,  and  the  ellipses  will  soon  take 
the  appearance  of  perfect  circles. 

Fig.  37  represents  a  ring  drawn  in  semi-diagonal  perspec- 
tive, and  in  the  same  position  as  shown  by  the  ellipse  on 
the  right-hand  side  of  the  cube,  Fig.  36.  The  inner  diam- 
eter of  the  ring,  as  measured  either  by  the  line  v  w  or  1  2. 
is  forty-eight  inches,  the  same  as  the  ellipse  1  v  2  w  of  Fig. 
3G.  The  thickness  of  the  ring  is  two  inches,  and  its  width 
four  inches. 

As  the  inner  front  ellipse  is  drawn  within  the  semi-diag- 
onal square,  (7  B  E  F,  so  the  outer  elliptic  circumference  of 
the  front  face  of  the  ring  must  be  drawn  within  a  semi-di- 
agonal square  two  inches  larger,  in  every  direction,  than 
the  inner  square,  as  shown  by  the  surrounding  dotted  semi- 
diagonal  square. 

The  true  width  of  the  ring  is  marked  out  by  extending 
the  points  of  the  compasses  two  spaces,  and  laying  off  that 


176  INDUSTRIAL   DRAWING.  [BOOK   NO.  IV. 

distance  horizontally  to  the  left,  from  the  right-hand  por- 
tion of  the  inner  front  ellipse,  and  also  from  the  left-hand 
portion  of  the  outer  front  ellipse.  Thus  the  horizontal 
measures  3  4, 5  6,  7  8,  9  10,  etc.,  must  each  be  equal  to  two 
spaces,  if  the  ellipse  be  accurately  drawn. 

Fig.  38  represents  a  pointed  arch  drawn  in  semi-diagonal 
perspective.  The  opening,  or  span  of  the  arch,  1  2,  is  forty 
inches;  and  the  ^7ertical  height,  3  4,  is  also  forty  inches — 
although  the  arch  proper  is  fourteen  inches  less  in  height, 
as  the  walls  do  not  begin  to  converge  until  they  reach  a 
height  of  fourteen  inches  from  the  base. 

If  this  arch  were  to  be  drawn  from  the  given  measure- 
ments, it  should  first  be  drawn  within  a  rectangular  square 
fronting  the  spectator.  It  could  then  be  changed  with  ac- 
curacy to  a  semi-diagonal  view,  according  to  the  method  ex- 
plained for  transferring  the  measures  of  Fig.  26  to  Fig.  27. 

Or,  if  the  nearer  inner  curve,  1  4-,  he  drawn  by  the  eye,  ac- 
cording to  the  judgment,  the  farther  inner  curve  may  be 
drawn  to  correspond  to  it.  Then  the  outer  curves,  6  5  and 
5  7,  must  be  made  to  correspond  to  the  inner  curves ;  and 
as  1 6  or  2  7  is  a  measure  of  four  inches,  so  the  point  5  must 
be  taken  four  inches  above  the  point  4-  -As  the  depth  of 
the  arch,  6  8  or  2  10,  is  six  inches,  so  the  depth,  wherever 
measured  on  a  horizontal  line,  as  a  b,  c  d,  11  12,  9  5,  etc., 
must  be  six  inches. 

Fig.  39  is  a  cylinder  one  foot  in  diameter,  and  seven  feet 
four  inches,  or  eighty-eight  inches,  in  length.  Its  axis  is  a  b. 
At  each  end  of  the  cylinder  is  a  projecting  tenon  twelve 
inches  long,  eight  inches  wide,  and  four  inches  in  thickness ; 
and  longitudinally  through  the  cylinder  is  a  mortise  three 
inches  wide,  extending  to  within  four  inches  of  the  ends, 
and  coinciding  in  direction  with  the  width  of  the  tenons. 

PROBLEMS   FOR   PRACTICE. 

1.  Draw  a  pointed  arch,  similar  to  Fig.  38,  first  in  rectangular  perspec- 
tive, and  then  in  semi-diagonal  perspective,  of  the  following  dimensions : 
Span  of  arch,  forty-eight  inches ;  height  of  opening,  forty-eight  inches ; 
thickness  of  \valls,  eight  inches ;  depth  of  arch,  twelve  inches.  [In  the 
first  drawing,  while  the  front  view  of  the  arch  is  to  be  on  a  rectangular  basis, 
and  therefore  in  its  relative  proportions  throughout,  the  side  view  is  to  be 


CABINET   PERSPECTIVE — MISCELLANEOUS.  177 

in  semi-diagonal  perspective— just  as  the  side  view  of  Fig.  33  is  in  semi-di- 
agonal perspective.] 

2.  Draw  a  partial  cylinder,  somewhat  similar  to  Fig.  30,  but  of  the  fol- 
lowing dimensions :  Length  of  cylinder,  eight  feet ;  diameter,  sixteen 
inches  ;  tenons,  sixteen  inches  in  length,  otherwise  the  same  as  in  Fig.  39  ; 
upper  side  of  the  cylinder  to  be  taken  off  horizontally,  even  with  the  top  of 
the  tenons  ;  and  a  mortise,  four  inches  wide  and  eighty  inches  in  length,  to 
extend  longitudinally  through  the  cylinder,  equidistant  from  the  two  ends, 
and  coinciding  in  direction  with  the  width  of  the  tenons,  as  in  Fig.  39. 


PAGE  ELEVEN.— SCALE  OF  FOUR  INCHES  TO  A  SPACE. 

Fig.  40.  According  to  the  scale  of  measurement  adopted, 
Fig.  40  represents  a  small  building,  fifteen  feet  long,  and 
ten  feet  eight  inches  wide,  with  posts  nine  feet  four  inches 
high.  The  four  corner  posts  and  sills  are  eight  inches  square 
at  the  ends ;  the  two  middle  posts  are  four  inches  square, 
and  the  central  cross-sill  is  four  by  eight  inches.  The  plates 
on  which  the  rafters  rest  are  four  by  eight  inches,  while  the 
rafters  are  four  inches  square.  The  rafters  are  placed  at 
what  is  called  half  pitch — the  height,  2  7,  being  equal  to 
half  the  span,  8  9;  or,  as  the  vertical  line  2  7  is  equal  to 
2  #,  the  rafters  are  at  an  angle  of  forty-five  degrees. 

Observe  that  the  lower  ends  of  the  rafters  come  down  at 
equal  distances  on  both  sides  below  the  plate,  as  indicated 
by  the  line  5  6;  and  that  the  ends  are  sawed  off  horizon- 
tally. The  rafters  are  twelve  inches  apart,  or  sixteen  inches 
between  their  central  lines. 

Observe  that  the  braces  are  all  of  the  same  length — forty 
inches,  inside  measure — and  that  they  are  placed  flush  with 
the  outside  of  the  frame.  Thus  the  inside  of  the  top  of  the 
extreme  brace  at  the  right,  at  s,  is  twenty-four  inches  from 
the  corner  t;  and  the  inside  of  the  bottom  of  the  brace,  at 
v,  is  thirty-two  inches  below  the  corner  t.  Now,  as  s  t  v  is 
a  right  angle,  s  v  is  the  hypothenuse ;  and  if  we  add  togeth- 
er the  squares  of  s  t  and  t  v,  and  extract  the  square  root  of 
the  sum,  we  shall  find  that  the  length  of  s  v  is  forty  inches. 
All  the  braces  are  arranged  in  like  manner.  In  all  cabinet 
work,  and  in  buildings,  braces  are  generally  arranged  in  the 
proportions  of  three  measures  for  one  side  of  the  triangle 
and  four  for  the  other;  and  then  five  will  be  the  measure 

H2 


178  INDUSTRIAL   DBA  WING.  ^BOOK   NO.  IV. 

for  the  hypothenuse,  whether  the  measure  be  in  inches  or 
in  feet. 

The  only  difficulty  in  drawings  similar  to  Fig.  40  con- 
sists in  placing  the  braces  in  their  correct  positions,  accord- 
ing to  the  measurements  assigned  to  them  ;  and,  in  order  to 
arrange  them  accurately,  it  is  evident  that  we  must  first 
find  those  corners  which  are  hidden  from  view,  as  shown 
in  Figs.  31  and  32.  The  following  problems  will  aid  in 
elucidating  the  principles  which  govern  the  drawing  of 
braces  in  the  various  positions  in  which  they  usually  occur 
in  a  building. 

PROBLEMS    FOR    PRACTICE. 

[In  these  problems  the  posts  are  to  be  sixteen  inches  square  at  the  ends; 
and  the  plates  or  cross-beams  are  to  be  four  by  sixteen  inches  at  the  ends. 
Referring  for  illustration  to  the  brace  at  A,  the  top  of  each  brace  (as  s)  is 
to  be  forty-eight  inches,  on  the  inside,  from  the  corner  (as  t)  where  the 
plate  joins  the  post ;  and  the  bottom  of  each  brace,  on  the  inside  (as  at  v\ 
is  to  be  sixty-four  inches  below  the  corner  (as  /)  where  the  plate  joins  the 
post.  The  inside  length  of  each  brace  will  then  be  eighty  inches.  The 
braces  are  to  be  twice  the  size  of  timber  shown  in  Fig.  40,  and  to  be  placed 
flush  with  the  outside  of  the  frame.  In  each  case,  let  the  outlines  of  the 
concealed  end  of  the  brace  be  dotted,  as  in  Figs.  31  and  32.] 

1 .  Draw  a  brace  of  the  given  dimensions  connecting  a  post  and  plate,  as 
in  the  corner  A.     Only  a  sufficient  length  of  post  and  plate  may  be  drawn 
to  show  the  plate  to  advantage. 

2.  Draw  a  brace  for  a  corner  corresponding  to  B. 

3.  Draw  a  brace  for  a  corner  corresponding  to  C. 

4.  Draw  braces  for  corners  corresponding  to  D  and  //. 

5.  Draw  a  square  frame-work  of  two  upright  posts,  and  plate,  and  sill, 
corresponding  to  the  left-hand  end  of  the  building,  Fig.  40,  and  put  a  brace 
of  the  given  dimensions,  and  flush  with  the  outside,  in  each  of  the  four  cor- 
ners of  the  frame. 

G.  Draw  a  brace  for  a  corner  corresponding  to  G,  and  also  one  for  a  cor- 
ner corresponding  to  7,  below  G. 

1.  Draw  braces  of  the  given  dimensions  for  corners  corresponding  to  the 
four  corners  embraced  by  the  sills  of  the  building,  and  let  the  braces  be 
flush  with  the  tops  of  the  sills.  .  Let  the  sills  be  sixteen  inches  square  at 
the  ends. 


CABINET   PERSPECTIVE MISCELLANEOUS.  179 

V.  SHADOWS  IN  CABINET  PERSPECTIVE. 
As  cabinet  perspective  is  designed  for  the  artisan  rather 
than  the  artist,  we  have  thus  far  striven  for  no  effect  in  our 
drawings  beyond  what  is  requisite  to  convey,  in  as  simple  a 
manner  as  possible,  correct  ideas  of  the  forms  and  dimen- 
sions of  objects.  To  this  end  such  simple  methods  of  plain 
shading  have  been  introduced  as  will  most  readily  distin- 
guish one  surface  from  another,  while  no  attention  has  been 
given  to  the  shadows  cast  by  objects.  But  where  it  is  de- 
sired to  give  greater  artistic  effect  to  cabinet  drawings, 
the  system  of  perspective  here  adopted  wrill  enable  the 
draughtsman  to  define  the  outlines  of  shadows  with  the 
greatest  ease,  and  with  a  degree  of  mathematical  accuracy 
hitherto  unattainable.'  On  page  12  we  give  a  few  illustra- 
tions of  the  principles  which  are  applicable  to  this  subject. 

PAGE  TWELVE.— SCALE  OF  SIX  INCHES  TO  A  SPACE. 

The  objects  here  represented  are  supposed  to  stand  on  a 
horizontal  plane,  which  may  be  understood  to  be  the  level 
surface  of  the  earth ;  and  the  spectator  is  supposed  to  be 
looking  down  upon  them  from  above,  and  at  the  right,  and 
from  a  northerly  direction.  Hence  the  left-hand  side  of  the 
paper  is  east,  the  upper  side  is  south,  the  right-hand  west, 
and  the  lower  side  is  north — as  indicated  by  the  large  cap- 
itals, K,  S.,  TV.,  N. 

Fig.  41.  This  figure  represents  a  square  vertical  pillar 
standing  upon  the  level  surface  of  the  earth,  while  the  sun, 
elevated  at  some  distance  above  the  horizon,  and  shining 
upon  the  pillar  from  a  southeasterly  direction,  causes  the  pil- 
lar to  cast  a  shadow  on  the  earth,  as  shown  in  the  drawing. 

The  sun  is  so  far  distant  from  the  earth  that  the  rays  of 
light  coming  from  it  may  be  regarded  as  parallel.  Suppose 
that  the  rays  of  light  come  from  the  southeast  in  a  semi- 
diagonal  direction,  as  indicated  by  the  arrows,  or,  c,  e;  and 
that  the  ray  #,  just  touching  the  corner  1,  strikes  the  earth 
at  the  point  9.  Then  it  is  evident  that  all  rays,  such  as 
g  h,  ij,  etc.,  that  strike  the  corner  line  1 2,  will  project  shad- 
ows upon  the  earth  between  the  points  £  and  9;  and  hence 


ISO  INDUSTRIAL   DRAWING.  [BOOK    NO.  IY. 

the  line  2  P,  which  indicates  a  line  of  shadow  drawn  on  the 
level  surface  of  the  earth,  will  be  the  shadow  of  the  vertical 
corner  line  1  2.  This  line  of  shadow,  2  9,  although  hori- 
zontal, will  diverge  away  from  the  line  2  4,  and  also  be 
lengthened,  just  in  proportion  to  the  southerly  direction  of 
the  sun,  and  its  nearness  to  the  eastern  horizon.  If  the  sun 
were  directly  east  of  the  line  1  2,  and  on  the  horizon,  its 
shadow  would  be  extended  indefinitely  from  2  in  the  direc- 
tion of  2  4>  but  the  shadow  would  become  shorter  and 
shorter  as  the  sun  rose  vertically  toward  the  zenith;  and 
when  at  the  zenith  the  shadow  of  1  2  would  be  contracted 
to  a  single  point  at  2. 

Hence  two  lines  are  required  to  define  the  shadow  cast 
by  the  vertical  line  12:  the  one,  a  9,  called  the  ray-lute,  giv- 
ing the  direction  of  the  ray  of  light  that  barely  touches 
the  corner  !•  and  the  other,  called  the  shadow-line,  drawn 
through  the  corner  2,  and  intersecting  the  ray-line  at  9. 

Hence,  to  find  the  shadow  cast  on  a  horizontal  plane  by 
any  given  vertical  line,  or  by  any  given  point: 

RULE  I. — Draw  a  BAY-LINE,  indicating  the  direction  of 
the  surfs  rays,  from  the  top  of  the  given  line  to  the  horizontal 
plane  which  is  on  a  level  with  the  lower  end  of  the  given  line; 
then  draw  a  SHADOW-LINE  from  the  lower  end  of  the  given 
line  to  the  point  where  the  ray-line  strikes  the  horizontal 
plane.  The  shadow-line  thus  drawn  will  be  the  shadow  of 
the  given  line. 

The  shadow  cast  by  any  given  POINT  on  a  horizontal  plane 
may  be  found  by  first  drawing  a  vertical  line  from  it  to  the 
horizontal  plane,  and  then  finding  the  shadow  of  the  given 
line,  as  before.  The  shadow  cast  by  any  required  point  in  the 
given  line  may  thus  be  obtained. 

Following  out  the  principles  of  light  and  shade  in  con- 
nection with  this  rule,  it  will  be  seen,  as  before  shown,  that 
2  9  is  the  shadow  of  the  vertical  line  1  2,  and  that  9  is  the 
shadow-point  of  the  point  1. 

In  a  similar  manner  it  is  found  that  the  line  4  8  would  bo 
the  line  of  shadow  cast  by  the  vertical  corner  line  3  J>,  if  all 
the  pillar  except  its  corner  line  3  4  were  transparent;  also 
that  6  7  is  the  line  of  the  shadow  cast  bv  the  corner  line 


CABINET    PERSPECTIVE MISCELLANEOUS.  181 

5  6.  The  line  8  7  must  therefore  be  the  line  of  shadow  cast 
by  the  line  35.  If  5  10  could  cast  a  shadow,  its  shadow 
would  be  the  line  7  11,  parallel  to  and  equal  in  length  to 
5  10;  and  if  10  1  could  cast  a  shadow,  its  shadow  would  be 
the  line  11  9,  parallel  to  and  equal  in  length  to  10  1. 

The  following  Rule  may  also  be  deduced  from  Fig.  41. 

RULE  II. — Every  horizontal  line  casts  a  shadow,  on  a 
horizontal  surface,  parallel  to  itself;  and  the  shadow  has  the 
same  representative  length  as  the  line  casting  the  shadow. 
Thus  9  8  is  parallel  to  and  equal  to  1  3;  8  7  to  3  5;  7  11 
to  5  10;  11  9  to  10  1,  etc. 

Fig.  42.  In  this  figure  the  horizontal  lines  of  shadow  cast 
on  the  ground  by  vertical  lines  are  represented  as  bearing 
in  the  direction  of  southwestern  four -space  diagonals,  as 
shown  by  the  course  of  the  arrows  «,  b,  c,  d,e;  and  the  rays 
of  light  as  coming  diagonally  downward,  really  from  the 
northeast,  as  indicated  by  the  arrows  f,g,  h,  i,j,  although 
they  seem  to  come  from  the  southeast.  Here  the  north 
and  east  sides  of  the  objects  are  in  the  light,  and  the  south 
and  west  sides  in  shadow,  as  they  would  be  if  the  objects 
were  south  of  the  equator,  and  the  sun  had  risen  midway 
toward  the  zenith. 

Observe,  here,  that  4  m  is  the  line  of  shadow  cast  by  3  4; 
m  n  the  line  of  shadow  cast  by  3  1 ;  and  n  v  a  part  of  the 
line  of  shadow  cast  by  1  t;  and  that  2  n  would  be  the 
shadow  cast  by  1  2,  if  1  2  could  cast  a  shadow.  Also,  6  13 
is  the  line  of  shadow  cast  by  the  corner  line  5  6;  14  15,  the 
shadow  cast  by  8  7 ;  15  16,  the  shadow  cast  by  7  9;  and 

16  17,  the  shadow  cast  by  9  11.     The  farther  vertical  corner 
line  of  the  pillar,  represented  by  the  dotted  line  11  12,  also 
casts  a  shadow  on  the  top  of  the  platform,  which  shadow  is 
represented  by  the  line  12  17;  but  only  a  small  portion, 

17  x,  of  this  line  of  shadow  is  visible.      Observe,  here,  the 
strict  application  of  Rule  II.,  as  to  the  direction  and  length 
of  the  shadows  cast  by  horizontal  lines.     Let  the  pupil  ex- 
plain the  method  of  finding  the  shadow  cast  by  any  given 
line  in  Fig.  42,  or  by  any  given  point  in  the  structure. 

Fig.  43.  In  this  figure  the  north  and  west  sides  of  the 
objects  are  in  shadow;  the  horizontal  lines  of  shadow  cast 


182  INDUSTRIAL   DRAWING.  [BOOK   NO.  IV. 

on  the  ground  by  vertical  lines  are  represented  as  bearing 
in  the  direction  of  northwestern  three-space  diagonals,  as 
shown  by  the  arrows  #,/",/*,&/  and  the  rays  of  light  as 
coming  semi-diagonally  downward  from  the  southeast,  as 
indicated  by  the  arrows  r,  s,  t,v,  etc.  The  important  point 
to  be  noticed  in  this  figure  is  that  the  square  pillar  casts 
its  shadow  beyond  the  platform,  and  beyond  the  shadow 
of  the  platform  also. 

The  shadows  cast  by  the  two  visible  sides  of  the  plat- 
form are  easily  obtained.  The  shadow  2  3,  of  the  vertical 
corner  line  1 2,  is  obtained  in  the  same  manner  as  the  shadow 
of  the  corresponding  corner  line  of  Fig.  41.  The  shadow 
that  would  be  cast  by  1  c  on  the  horizontal  surface  of  the 
earth  must  be  parallel  to  and  equal  in  apparent  length  to 
1  c.  (See  Rule  II.)  Now,  if  the  platform  did  not  intercept 
a  portion  of  this  shadow,  the  shadow  of  1  c  would  be  the 
line  5  7;  but  the  platform  intercepts  a  portion  equal  to  3  4 
— that  is,  the  shadow  of  that  part  of  the  line  included  be- 
tween the  points  1  and  b;  and  it  is  the  shadow  of  the  part 
b  c  only  that  passes  beyond  the  platform,  and  shows  itself 
on  the  ground  in  the  line  6  7.  The  two  shadow-lines  3  4 
and  6  7. must  therefore  be  equal  to  1  c. 

Having  now  the  point  7  as  the  point  of  shadow  cast  by 
the  corner  c,  we  know  (Rule  II.)  that  7  #,  drawn  parallel 
to  and  equal  to  c  d,  must  be  the  line  of  shadow  of  c  d  ;  and 
hence  8  will  be  the  point  of  shadow  of  the  corner  d.  But 
this  point  of  shadow  may  also  be  found  by  the  general 
rule,  in  the  following  manner  (see  Rule  I.) :  Extend  the 
vertical  corner  line  d  i  down  to  m — that  is,  to  the  level  of 
the  earth  on  which  the  platform  rests ;  and  then  the 
problem  becomes  one  to  find  the  shadow  cast  on  the  earth 
by  the  line  d  m.  From  m  draw  a  three-space  diagonal  line 
of  shadow  in  the  direction  m  8;  and  through  d  draw  a  two- 
space  diagonal  ray-line,  which,  at  its  intersection  with  the 
former  line,  will  give  the  point  8. 

We  next  wish  to  find  the  shadow  cast  by  the  vertical 
corner  line  d  i.  It  is  evident  that  the  shadow  that  would 
be  cast  on  the  ground  by  the  whole  line  d  m  would  be  the 
line  m  8;  but  9  8  is  the  only  part  of  this  shadow-line  that 


CABINET   PERSPECTIVE MISCELLANEOUS.  183 

is  not  intercepted  by  the  platform ;  and  9  8  is  the  shadow 
cast  by  g  d.  Kow  the  shadow  cast  by  the  portion  g  i  is 
evidently  i  n, which  latter  line  is  found  by  drawing  a  three- 
space  diagonal  line  of  shadow  through  the  corner  «,  and  in- 
tersecting it  by  the  two-space  diagonal  ray-line  g  n.  This 
completes  the  outlines  of  shadow  cast  by  the  platform  and 
the  pillar. 

Fig.  44  represents  two  rectangular  blocks,  A  and  B,  in 
vertical  position,  and  standing  at  right  angles  to  one  an- 
other upon  the  horizontal  surface  of  the  earth.  The  sun  is 
in  the  southwest,  and  at  such  an  elevation  that  its  rays  pass 
downward  in  the  direction  of  semi-diagonals,  as  indicated 
by  the  arrows  s,tyu,v,x;  while  the  shadows  cast  by  vertical 
lines  are  in  the  direction  of  horizontal  semi-diagonals,  as 
indicated  by  the  arrows  c,c?,/.  Both  blocks  cast  shadows 
upon  the  earth;  while  the  block  B  casts  a  shadow,  ad- 
ditionally, upon  a  part  of  the  vertical  surface  of  the  block  A. 

In  accordance  with  principles  already  explained,  the  line 
2  g  is  the  shadow  cast  by  the  vertical  corner  line  1 2.  Hav- 
ing g  as  the  point  of  shadow  cast  by  the  comer  .7,  we  know 
that  the  shadow  cast  on  the  earth  by  the  line  1  7  would 
extend  to  the  left  from  <?,  parallel  to  1  7,  and  of  a  length 
equal  to  1  7.  But  this  line  of  shadow  is  intercepted  at  9 
by  the  vertical  surface  of  the  block  A  ;  and  it  is  evident 
that  g  9  is  the  shadow  of  1  10  only.  Where,  then,  does  the 
part  10  7  cast  its  shadow  ?  As  the  point  10  casts  its  shadow 
at  9,  and  as  the  shadow  of  7  is  at  7  itself,  on  the  vertical 
surface  of  A,  it  follows  that  the  straight  line  10  7  must  cast 
its  shadow  in  a  straight  line  from  9  to  7,  from  which  we  de- 
duce the  following  rule,  applicable  to  all  similar  cases. 

RULE  HI. —  The  two  points  of  shadow  cast  on  a  plane 
surface  by  the  extremities  of  a  straight  line  being  given,  the 
shadow  of  the  line  will  be  in  a  straight  line  connecting  the 
two  given  points  of  shadow. 

The  additional  lines  of  shadow  in  Fig.  44  present  no  diffi- 
culties— the  line  4  5  being  the  shadow  of  3  4,  and  5  6  being 
the  shadow  of  3  i.  The  line  i  r  casts  a  shadow,  and  a  part 
of  it  is  out  of  view, 'on  the  farther  or  eastern  side  of  the 
block  A.  But  the  shadow  of  the  corner  i  is  at  6;  and  the 


184  INDUSTRIAL   DRAWING.  [BOOK    NO.  IV. 

shadow  of  i  r  must  extend  from  6  to  m,  parallel  to  i  r,  find 
equal  in  length  to  i  r  (Rule  II.).  Or  the  point  of  shadov.- 
cast  by  the  corner  r  may  be  found  by  the  regular  method, 
as  follows:  As  n  is  the  farther  but  invisible  corner  of  the 
block,  directly  below  r,  if  we  draw  a  horizontal  two-space 
line  of  shadow  through  n,  and  a  two-space  diagonal  ray-line 
through  r,  their  point  of  intersection  at  m  will  be  the  point 
of  shadow  cast  by  the  corner  r. 

Fig.  45.  In  Fig.  45  the  ray-lines,  and  the  shadows  of  verti- 
cal lines,  are  the  same  as  in  the  preceding  figure ;  and  the 
two  blocks  C  and  D  are  similar  in  position  to  the  blocks  A 
and  JB  of  Fig.  44 ;  but  they  here  rest  on  a  horizontal  plat- 
form. 

In  this  figure  the  lines  of  shadow  b  a  and  a  1  are  in  all 
respects  similar  to  the  lines  of  shadow  g  9  and  9  7  in  Fi<r. 
44 — except  that  only  that  part  ofal  (viz.,  a  £),  which  touch- 
es the  vertical  side  of  (7,  is  a  shadow-line.  Here  b  a  is  the 
shadow  of  8  r;  a  6  is  the  shadow  of  r  t;  and  6  3  is  the 
shadow  of  t  1. 

To  find,  independently,  the  point  of  shadow  cast  by  the 
corner  1,  through  1  draw  the  ray-line  1  n;  and  through  2, 
the  point  where  the  corner  line  1  10  comes  in  contact  with 
the  upper  surface  of  (7,  draw  the  horizontal  two-space  line 
of  shadow  2  fn  ;  and  the  intersection  of  these  two  lines,  at 
8,  will  be  the  point  of  shadow  of  the  corner  1.  Draw  3  4 
equal  and  parallel  to  1  7,  and  3  4  will  be  the  line  of  shadow 
of  1  7.  Now,  as  4  is  the  point  of  shadow  of  the  corner  7, 
and  5  is  that  point  in  the  farther  vertical  corner  line  of  the 
block  directly  below  7,  and  level  with  the  upper  surface  of 
C,  it  follows  that  ^  5  is  the  line  of  shadow  of  7  5 — that  part 
of  the  farther  vertical  corner  line  that  is  above  the  block  C. 
Only  a  small  part  of  this  shadow-line,  4  5,  is  visible. 

The  shadows  cast  on  the  ground  by  the  platform,  and  by 
the  end  of  the  block  (7,  require  no  farther  explanation  than 
may  be  obtained  from  the  drawing  itself. 

Fig.  46  is  drawn  in  semi-diagonal  cabinet  perspective; 
and  it  will  be  seen,  from  this  figure,  that  the  same  principles 
and  rules  of  shadow  apply  to  semi-diagonal  as  to  diagonal 
cabinet  perspective. 


CABINET   PERSPECTIVE MISCELLANEOUS.  185 

In  this  drawing  the  sun  is  supposed  to  be  southwest  from 
the  spectator,  half  way  between  the  zenith  and  the  horizon ; 
and  hence  the  ray-lines  are  in  the  direction  of  diagonals,  as 
indicated  by  the  arrows  a,  b,  c,  etc. ;  while  we  assume  that 
the  lines  of  shadow  cast  by  the  vertical  lines  on  horizontal 
surfaces  are  in  the  direction  of  three-space  diagonals,  as  in- 
dicated by  the  arrows  e,/,  g,  etc. 

From  the  principles  already  explained,  it  is  easy  to  find 
the  shadows  cast  on  the  ground  by  the  lines  forming  the 
ends  of  the  steps;  and  the  shadows  cast  by  the  block  I^on 
the  top  of  the  first  step,  and  on  the  riser  of  the  second,  are 
in  all  respects  like  the  shadows  cast,  in  Fig.  44,  by  the  block 
IB  on  the  ground,  and  on  the  vertical  side  of  the  block  A. 
It  is  evident,  also,  that  the  line  1  2  (a  part  of  the  vertical 
corner  line  6  2)  casts  its  shadow  on  the  top  of  the  second 
step,  from  2  to  3. 

Now  the  point  1  casts  its  shadow  at  the  point  3 ;  and  we 
find,  in  the  following  manner,  that  the  point  4  casts  its  shad- 
ow at  the  point  5.  Take  the  point  m  in  the  line  6  2,  on  a 
level  with  the  top  of  the  third  step,  and  through  it  draw  the 
indefinite  shadow-line  in  the  direction  m  5.  Through  4  draw 
a  diagonal  ray-line,  and  the  point  5,  at  which  it  intersects 
the  line  drawn  through  m,  will  be  the  point  of  shadow  cast 
by  the  point  4->  in  accordance  with  Rule  I.  But  5  3  is  par- 
allel to  and  equal  in  length  to  4  1 — from  which  we  derive 
the  following  rule. 

RULE  IV. — The  shadow  cast,  by  a  vertical  line,  on  a  ver- 
tical plane  surface,  is  parallel  to  the  vertical  line  itself;  and, 
if  the  vertical  plane  surface  have  sufficient  extent,  the  shadow 
will  have  the  same  length  as  the  line  casting  the  shadow. 

We  next  wish  to  find  the  shadow  cast  by  the  part  4  6  of 
the  line  6  2;  and  to  this  end  we  first  find,  in  accordance 
with  Rule  I.,  the  point  of  shadow  cast  by  the  corner  6. 
Thus :  we  take  the  point  m,  in  the  line  6  2,  on  a  level  with 
the  top  of  the  third  step,  and  through  m  draw  a  line  in  the 
direction  (m  7)  of  the  shadows  for  vertical  lines ;  and  then 
through  6  draw  a  ray-line,  which  we  find  intersects  the  for- 
mer line  at  7,  thus  showing  that  7  is  the  point  of  shadow 
cast  by  the  corner  6. 


186  INDUSTRIAL   DRAWING.  [BOOK   NO.  IV. 

We  are  next  to  find  the  shadows  cast  by  the  line  6  h. 
Referring  to  Rule  II.,  we  find,  as  7  is  the  shadow-point  of  6, 
so  7  9  is  the  shadow  of  6  8. 

Also,  referring  to  Fig.  44,  we  find,  on  the  principles  there 
explained,  that  as  7  9,  in  Fig.  44,  is  the  shadow  cast  by 
7 10;  so,  in  Fig.  46, 9  11  is  the  shadow  cast  by  8  10.  Or,  by 
taking  the  point  s,  vertically  below  10,  and  on  a  level  with 
the  top  of  the  fourth  step,  we  may  show  by  the  triangle, 
10  s  11,  that  the  point  11  is  the  shadow  of  the  point  10 
(Rule  I). 

Again,  by  Rule  II,  11  13  is  the  shadow  of  10  12.  Also, 
according  to  the  principle  just  referred  to  in  Fig.  44, 13  15 
is  the  shadow  of  12  14.  And  again,  by  Rule  II.,  15  17  is  the 
shadow  of  14  16. 

We  have  next  to  find  the  shadow  cast  by  that  portion  of 
the  line  embraced  between  16  and  h.  As  16  is  the  farthest 
point  in  the  line  6  h  that  casts  its  shadow  on  the  steps, 
therefore  16  h  must  evidently  cast  its  shadow  beyond  the 
steps,  on  the  ground.  Let  us,  then,  find  the  point  of  shadow 
cast  by  the  corner  h. 

Take  the  point  v,  fifteen  spaces  vertically  below  h,  on 
the  ground-level ;  draw  the  three-space  diagonal  line  of 
shadow  through  v,  and  the  diagonal  ray-line  through  h; 
and  their  point  of  intersection,  at  j,  will  be  the  point  of 
shadow  cast  on  the  level  ground  by  the  corner  h.  Then 
j  i,  drawn  parallel  and  equal  to  h  16,  will  be  the  line  of 
shadow  of  h  16.  The  other  portion  of  h  6,  at  the  right  of 
16,  casts  its  shadow  on  the  steps,  as  already  shown ;  and 
at  the  point  i  its  shadow  is  lost  in  the  shadow  of  the 
steps. 

The  line  j  x,  equal  and  parallel  to  h  t,  is  the  shadow  of 
h  t;  and  x  u  would  be  the  shadow  of  the  farther  vertical 
but  invisible  corner  line  t  u;  but  only  the  portion  x  z,  of 
this  shadow-line,  can  be  seen.  Observe,  also,  that  as  the 
ray-line  passing  through  16  just  touches  the  point  17,  and 
strikes  the  ground  at  i,  and  as  r  is  the  point  of  shadow  cast 
by  p,  therefore  i  r,  equal  and  parallel  to  17 p,  must  be  the 
shadow  cast  on  the  ground  by  17  p.  So  accurately  do  all 
portions  of  the  shadows  harmonize  with  one  another,  and 


CABINET   PERSPECTIVE MISCELLANEOUS.  187 

beautifully  illustrate  the  principles  of  shadows,  as  deduced 
from  this  system  of  drawing. 

The  shadows  cast  by  curvilinear  objects  may  be  defined 
with  equal  accuracy,  on  the  general  principles  already  ex- 
plained, with  some  modifications ;  but  we  have  not  space  to 
illustrate  the  subject  here. 


APPENDIX. 


ISOMETRICAL    DRAWING. 
I.  ELEMENTARY  PRINCIPLES. 


TSOMETRICAL  DsAwixG,  or  Isometrical  Perspective,  is  based 
upon  the  following  principles  :  If  a  cubical  block,  as  shown 
in  Fig.  1,  Plate  I.,  and  as  seen  shaded  in  Fig.  2,  be  supposed 
to  be  viewed  from  an  infinite  distance,  and  from  such  a  po- 
sition that  the  line  of  vision  shall  pass  through  the  upper 
and  nearer  corner,  1,  and  also  through  the  lower  farther 
corner,  the  three  visible  faces,  A,  B,  GT,  of  the  cube  will  ap- 
pear to  be  equal  in  measure,  the  one  to  the  other.  Any 
boundary-line  of  the  upper  surface,  A,  will  measure  the 
same  as  any  boundary-line  of  the  face  .Z?,  or  of  the  face  C. 
Thus  the  line  1  4  will  measure  the  same  as  the  line  1  6,  or 
4  5,  or  6  5,  or  1  ,?,  or  2  3,  etc.  And  any  measure  taken  on 
any  one  line  will  give  the  same  relative  distance  when  ap- 
plied to  any  other  line.  Hence  the  appropriateness  of  the 
term  Isometrical,  which  is  formed  from  two  Greek  words 
signifying  equal  in  measurement. 

The  isometrical  cube  is  based  upon  the  geometrical  prin- 
ciple for  inscribing  a  hexagon  in  a  circle.  Thus,  to  in- 
scribe a  hexagon  in  the  circle,  Fig.  1,  take  the  radius,  1  4, 
and,  beginning  at  2,  apply  it  six  times  to  the  circumference, 
and  it  will  give  the  points  2,  3,  4->  59  #>  7.  Join  these  points 
by  straight  lines,  and  we  shall  have  the  six  equal  sides  of 
a  regular  hexagon.  Connect  the  alternate  corners  of  the 
hexagon  with  the  central  point,  1,  and  retain  the  circumfer- 
ence of  the  circle,  and  we  shall  have  the  isometrical  cube  in- 
scribed in  a  circle. 


190  APPENDIX. 

In  the  isoraetrical  cube,  which  is  supposed  to  stand  upon 
a  horizontal  surface,  while  the  spectator  looks  down  upon  it 
diagonally,  each  of  the  lines  2  S  and  2  7  forms  an  angle  of 
sixty  degrees  with  the  vertical  line  2  1,  and  an  angle  of 
thirty  degrees  with  the  horizontal  line  8  9.  It  will  be  ob- 
served, also,  that  the  lines  1  4  and  6  5  are  parallel  to  2  3  ; 
1  6  and  4  5  parallel  to  27;  34  and  7  6  parallel  to  21.  In 
the  isometrical  cube,  therefore,  and  in  all  isoraetrical  draw- 
ing, there  are  only  three  kinds  of  true  isometrical  straight 
lines — vertical  Imes,  and  the  two  kinds  of  diagonals  as  seen 
in  Fig.  1.  But  unless  the  diagonal  lines  form  exact  angles 
of  sixty  degrees  with  the  vertical  lines,  the  drawings  made 
on  them  will  be  distorted ;  and  as  these  lines  can  not  be 
made  with  sufficient  accuracy  with  the  pen  or  pencil,  we 
have  had  them  accurately  engraved,  and  printed  in  red  ink 
on  drawing-paper.  By  the  aid  of  such  paper  all  difficulty 
in  making  accurate  isometrical  drawings  is  now  removed, 
as  the  ruling  is  a  perfect  guide  for  the  direction  of  all  the 
diagonal  lines ;  and  the  vertical  lines,  as  will  be  seen,  follow 
the  intersections  of  the  diagonals. 

Let  it  be  observed,  also,  that  the  diagonal  distance  from 
point  to  point  in  the  intersections  of  the  diagonal  lines  is 
precisely  the  same  as  the  vertical  distance  between  their  in- 
tersections. Thus,  in  Fig.  1,  the  five  diagonal  spaces  from 
1  to  4->  or  1  to  6,  or  2  to  7,  etc.,  measure  the  same  as  the  five 
vertical  spaces  from  1  to  2,  or  4  to  #,  or  6  to  7,  etc.  More- 
over, the  Isometrical  Drawing-Paper  is  so  ruled  as  to  cor- 
respond, in  measure,  to  the  ruling  of  the  Cabinet  Drawiny- 
Paper — what  is  called  a  space  in  the  one  corresponding  to 
a  space  in  the  other.  From  the  foregoing  explanation,  the 
pupil  who  is  familiar  with  the  principles  of  cabinet  perspec- 
tive will  have  little  difficulty  in  making  every  variety  of 
plane  isoraetrical  drawings. 


ISO1IETRICAL   DRAWING.  191 

II.  FIGURES    HAVING    PLANE    ANGLES. 
PLATE  I.— SCALE  OF  TWO  INCHES  TO  A  SPACE. 

According  to  the  scale  here  adopted,  Fig.  1  represents  the 
outlines  of  a  cube  of  ten  inches  to  a  side,  and  Fig.  2  is  the 
same,  shaded.  The  student  will  notice  the  difference  be- 
tween the  mode  of  measurement  here  adopted  and  that 
used  in  cabinet  perspective.  In  the  latter,  also,  one  face  of 
the  cube — the  front  vertical  face — would  be  drawn  in  its 
natural  proportions,  as  a  perfect  square. 

Fig.  3  represents  a  cube  of  ten  inches  to  a  side,  having 
rectangular  pieces  six  inches  square  and  two  inches  in  thick- 
ness cut  from  the  centres  of  its  three  visible  faces.  Let  the 
pupil  compare  this  drawing  with  that  of  Fig.  12,  page  1,  of 
Drawing-Book  No.  II.  The  cube  shows  to  excellent  advan- 
tage in  isometrical  drawing. 

Fig.  4  represents  an  inverted  frame  sixteen  inches  square, 
with  comer  posts  two  inches  square  and  six  inches  in  length. 

Fig.  5  is"  the  same  as  the  English  cross  bond  shown  in 
Fig.  38,  page  4,  of  Drawing-Book  No.  II.  The  two  figures 
illustrate,  very  happily,  the  two  methods  of  representation 
— cabinet  and  isometrical.  The  scale  adopted  being  the 
same  in  both  cases,  the  bricks  measure  the  same  in  both. 

Fig.  6  is  the  same  as  the  upper  part  of  Fig.  88,  page  11, 
of  Drawing-Book  No.  II.  A  figure  of  this  kind,  evidently, 
does  not  show  to  so  good  advantage  in  isometrical  as  in 
cabinet  drawing.  The  former  is  best  adapted  to  the  rep- 
resentation of  objects  whose  side  views  are  nearly  equal  in 
proportion. 

PROBLEMS   FOR   PRACTICE. 

"We  would  no\v  recommend  the  pupil  to  draw,  in  isometrical  perspective, 
all  the  figures  given  on  the  first  five  pages  of  Drawing-Book  No.  II.  Let 
him  take  the  measures  as  there  indicated  by  the  scale,  but  let  him  remem- 
ber that  a  diagonal  space  is  there  to  be  taken  as  twice  the  length  of  a  ver- 
tical or  horizontal  space,  while  in  isometrical  drawing  a  diagonal  space 
and  a  vertical  space  measure  the  same,  and  are  to  be  considered  of  equal 
length. 

The  pupil  would  do  well  to  draw  all  the  problems,  also,  connected  with 
these  first  five  pages. 


192  APPENDIX. 

PLATE  II.— SCALE  OF  ONE  FOOT  TO  A  SPACE. 

"We  have  here  adopted  a  scale  of  one  foot  to  a  space,  al- 
though any  scale  whatever,  that  is  most  convenient,  maybe 
used. 

Fig.  7  is  intended  to  represent  the  upper  part  of  a  pillar 
five  feet  square,  around  which  a  moulding  of  one  foot  pro- 
jection and  one  foot  in  height  is  to  be  placed,  even  with  the 
top.  The  shaded  portion  shows  the  attachment  of  the 
moulding  to  the  pillar. 

Fig.  8  shows  the  moulding  as  attached,  and  concealing 
from  view  a  portion  of  the  pillar  down  to  the  line  8  9  10. 
Hence  the  following  rule  : 

RULE. — Any  horizontal  rectangular  moulding  attached 
to  a  vertical  surface  obstructs  the  mew  of  that  surface  below 
the  moulding  to  an  extent  equal  to  the  extent  of  the  projection 
of  the  moulding. 

Fig.  9.  The  dotted  outline  represents  a  cubical  block  four 
feet  square,  while  the  shaded  portion  shows  a  "Wedge  cut 
from  it.  The  sides  of  the  wedge  bevel  off  equally  from  the 
sharp  edge  1  2,  inasmuch  as  the  lines  1  4  and  1  5  intersect 
the  base  line  4  &  a*  equal  distances  from  the  point  8. 

Fig.  10.  The  pillar  in  this  case  is  of  the  same  size  as  that 
seen  in  Fig.  8 ;  but  in  Fig.  10  the  moulding  is  cut  up  into 
three  cubical  blocks  on  a  side,  each  one  foot  square. 

Fig.  11  shows  how  triangular  blocks  attached  to  the  top 
of  a  column  may  be  represented.  The  dotted  continuations 
of  the  lines  of  the  farther  two  blocks  show  the  concealed 
points  on  the  column  toward  which  the  lines  are  to  be 
drawn. 

Fig.  12  represents  a  truncated  pyramid,  the  base  of  which 
is  surrounded  by  rectangular  mouldings.  Observe  that  the 
side  lines  of  the  pyramid  are  drawn  toward  the  point  x. 

Fig.  13.  Very  tall  four-sided  pillars,  gradually  tapering, 
and  having  a  flat  pyramid  at  the  summit,  as  at  A  and  .#, 
are  called  obelisks.  Observe  that  the  apex,  in  these  two 
obelisks,  is  in  the  central  vertical  line  of  the  pyramid. 

Fig.  14.  This  pyramid  has  a  rectangular  section,  one  foot 
in  depth,  cut  from  each  of  the  two  visible  sides  of  the  base. 


ISOMETRICAL   DRAWING.  193 

and  triangular  sections  cut  through  the  pyramidal  portion, 
so  that  all  except  the  four  edges  of  the  pyramid,  and  one 
foot  in  thickness  of  its  base  proper,  are  cut  away.  The  far- 
ther edge  of  the  pyramid  is  concealed  by  the  front  edge. 

PLATE  III.— SCALE  OF  TWO  FEET  TO  A  SPACE. 

Although  any  object  drawn  isometrically  is  supposed  to 
be  viewed  in  the  direction  of  the  diagonal  of  a  cube,  yet  we 
may  view  any  one  face  of  a  cube,  or  any  one  side  of  any 
rectangular  object,  from  four  different  positions,  and  at  the 
same  time  view  it  in  the  direction  of  some  one  of  the  diag- 
onals of  a  cube.  Thus : 

Fig.  15  represents  a  block  viewed  in  the  direction  of  the 
diagonal  that  passes  through  the  corner  1.  We  here  see 
the  top,  front,  and  right  side. 

Fig.  16  represents  the  same  block  viewed  in  the  direction 
of  the  diagonal  that  passes  through  the  corner  2.  We  here 
see  the  top,  front,  and  left  side. 

Fig.  17  represents  the  same  block  viewed  in  the  direction 
of  the  diagonal  that  passes  through  the  corner  4.  We  here 
see  the  bottom,  front,  and  right  side. 

Fig.  18  represents  the  same  block  viewed  in  the  direction 
of  the  diagonal  that  passes  through  the  corner  3.  We  here 
see  the  bottom,  front,  and  left  side. 

Figs.  15  and  16  are  viewed  from  above,  and  17  and  18 
from  below.  These  are  similar  to  tl.e  different  views  of  ob- 
jects in  cabinet  drawing,  as  represented  on  page  1  of  Draw- 
ing-Book No.  IV. 

Fig.  19  is  the  same  as  Fig.  56,  of  page  7,  in  Drawing-Book 
Xo.  II.,  although  the  designated  scales  are  different.  By 
adopting  the  same  scale,  the  figures  will  measure  alike. 

Fig.  20  represents  two  flights  of  steps,  ascending  in  dif- 
ferent directions,  and  leading  to  a  platform  seven  feet  in 
height.  As  each  step  rises  half  a  space — that  is,  one  foot, 
seven  steps  are  required  to  reach  the  platform. 

Fig.  21.  The  upper  roof,  A,  of  this  structure  is  evidently 
horizontal.  The  second  portion,  .Z?,  declines  downward 
from  the  horizontal,  as  represented  by  the  extent  to  which 
the  line  1  8  diverges  downward  from  the  diagonal  horizon- 

I 


194  APPENDIX. 

tal  line  1  2.  If  the  portion  J5  \Vere  horizontal,  and  of  equal 
width  on  both  sides,  the  line  6  3  would  extend  to  7,  and  the 
line  3  9  would  coincide  with  7  8. 

Again:  if  the  portion  C  were  vertical,  the  line  3  4  would 
coincide  in  appearance  with  the  line  3  5.  The  other  por- 
tions of  the  structure  require  no  explanation. 

PROBLEMS   FOE   PRACTICE. 

The  pupil  ought  now  to  find  no  difficulty  in  changing  all  the  figures  from 
page  6  to  page  11  inclusive,  and  Figs.  09,  100,  101,  and  102,  of  page  12, 
of  Drawing-Book  No.  II.,  into  isometrical  drawings.  If  he  think  this 
would  require  too  much  labor,  he  would  do  well  to  work  out  the  problems, 
at  least,  isometrically. 


III.  THE  DRAWING   OF  ISOMETRICAL  ANGLES. 

The  rectangular  ruling  on  the  upper  part  of  Plate  IV.  is 
designed  to  correspond  precisely  in  the  measure  of  its 
spaces — that  is,  in  the  distance  from  line  to  line,  measured 
on  the  lines — to  the  isometrical  ruling  on  the  lower  part  of 
the  plate.  So,  also,  the  ruling  on  the  "  Isometrical  Drawing- 
Paper"  corresponds,  in  like  manner,  to  the  ruling  on  the 
"  Cabinet  Drawing  -  Paper."  On  this  basis  we  have  con- 
structed a  "  Scale  of  Angles,"  w^hich  is  applicable  alike  to 
the  drawing  of  angles  on  both  isometrical  and  rectangular 
bases.  For  inasmuch  as  any  one  of  the  four  angles  of  a 
rectangular  square  may  be  divided  into  ninety  equal  de- 
grees, so  also  may  any  one  of  the  four  angles  of  a  corre- 
sponding isometrical  square  be  divided  into  equivalent  iso- 
metrical degrees,  isometrically  representing  the  angles  of 
the  rectangular  square.  Thus : 

Fig.  22.  Scale  of  one  foot  to  a  space. — In  the  geomet- 
rical square  A  ~B  C  Z>,  the  quarter -circle  B  D  is  divided, 
by  the  full  lines  which  diverge  from  A,  into  nine  equal 
parts,  each  part  representing  ten  degrees  at  the  corner  A. 
The  division  is  best  made  by  the  compasses,  in  the  follow- 
ing manner : 

From  D,  with  the  distance  D  A,  cut  the  curve  I>  D  at  sy 
and  from  J5,  with  the  same  distance,  cut  the  curve  at  t, 


ISOilETKICAL   DEAWIXG.  195 

The  curve  IB  D  will  thus  be  divided  into  three  equal  parts, 
representing  angles  of  thirty  degrees  each  at  the  point  A. 
Next  divide  each  of  these  parts,  by  the  compasses,  in  to  three 
equal  portions,  and  the  entire  curve  will  then  be  divided 
into  nine  equal  parts,  of  ten  degrees  each.  Through  these 
points  of  division  draw  lines  from  A,  and  extend  them  to 
the  sides  £  C  and  D  C  of  the  square.  Draw  a  dotted  line 
from  A  to  C,  and  the  angle  D  A  C  will  be  half  a  right 
angle — that  is,  an  angle  of  forty-five  degrees,  while  each  of 
the  angles  D  A  10, 10  A  20, 20  A  30,  etc.,  will  be  an  angle 
often  degrees. 

Within  the  larger  square,  A  B  CD,  we  may  count  twen- 
ty-five different  squares,  each  having  one  of  its  angles  at  A; 
and  on  the  two  sides  of  each  of  these  squares,  opposite  A, 
we  have  the  same  degrees  marked  off,  by  the  lines  diverg- 
ing from  A,  that  we  have  on  the  sides  B  C  and  D  C  of 
the  larger  square.  Thus  the  measure  8  p,  on  the  side  of  a 
square  of  eight  spaces,  measures  an  angle  of  twenty  degrees 
at  A,  as  truly  as  the  measure  D  20  measures  the  same  an- 
gle; and  8  r  measures  an  angle  of  forty-five  degrees,  just 
as  effectually  as  D  C  measures  the  same  angle. 

Now,  inasmuch  as  any  one  of  the  twenty-six  rectangular 
squares  that  may  here  be  designated  exactly  measures  an 
isometric  square  of  the  same  number  of  spaces  to  a  side, 
the  measures  of  angles  on  any  one  of  these  rectangular 
squares  may  be  used  to  lay  off  like  angles  on  a  correspond- 
ing isometric  square.  Thus : 

Fig.  23.  It  is  required  to  lay  off,  from  the  point  1  in  the 
line  1  2,  an  angle  of  ten  degrees.  As  the  lines  1  2  and  2  3 
are  two  sides  of  an  eight-space  isometric  square,  they  corre- 
spond to  the  two  lines  A  8  and  8  r  (in  Fig.  22),  two  sides 
of  an  eight-space  rectangular  square,  and  measure  the  same. 
From  the  point  8,  on  the  line  A  D,  take  the  distance  8  a, 
and  apply  it  to  the  isometric  square  on  the  line  from  2  to  3, 
and  mark  the  point  5.  A  line  drawn  from  5  to  1  will  then 
correspond  to  the  line  a  A  ;  and  the  isometric  angle  512 
will  correspond  to  the  angle  a  A  8,  and  will  represent  an 
angle  of  ten  degrees. 

If  from  the  point  4,  in  the  line  4  3,  of  the  isometric  square 


196  APPEXDIX. 

of  eight  spaces,  we  would  lay  off  an  angle  of  twenty  de- 
grees, lay  off  $  7  equal  to  8  p;  draw  a  line  from  7  to  4; 
and  the  angle  S  4  7  will  be  an  isometric  angle  of  twenty 
degrees,  the  same  as  8  A  p  is  an  angle  of  twenty  degrees. 

To  lay  off  an  angle  of  twenty  degrees  from  the  point  b 
in  the  line  b  d,  make  d  c  equal  to  S  p,  and  connect  c  b.  The 
angle  c  b  d  will  then  be  an  angle  of  twenty  degrees. 

To  lay  off  an  angle  of  forty  degrees  at  the  point  g  in  the 
line  g  A,  form  an  isometric  square,  as  g  h  k  n,  of  five  spaces, 
and  from  h  lay  off  h  i  equal  to  5  m  of  the  five-space  rectan- 
gular square,  and  connect  g  i.  Then  h  g  i  will  be  an  iso- 
metric angle  of  forty  degrees,  the  same  as  the  angle  5  A  m 
is  an  angle  of  forty  degrees. 

The  angles  laid  off  in  Figures  24  and  25  may  now  be  eas- 
ily drawn.  It  is  not  necessary,  in  any  case,  to  lay  off  a  full 
isometric  square  to  correspond  to  the  rectangular  square. 
It  is  sufficient  to  have  one  side  of  the  isometric  square,  and 
enough  of  the  other  side  to  receive  the  measure  from  the 
rectangular  square. 

If,  in  Fig.  19,  Plate  III.,  it  be  required  to  make  2  1  3  a 
certain  angle,  the  angle  may  be  laid  off  in  the  manner 
just  illustrated.  The  same  with  any  other  angle  which  it 
may  be  required  to  draw  on  any  isometrical  square.  So 
also,  in  Fig.  21,  if  it  be  known  what  angle  the  line  3  1  forms, 
in  the  real  object,  with  the  horizontal  line  1  2,  or  1  7,  the 
angle  may  be  laid  off  from  the  scale,  by  considering  that 
1  2  or  1  7  corresponds  to  a  portion  of  the  line  A  C  of  the 
scale.  The  angle  7  1  10  is  then  an  isometric  angle  of  forty- 
five  degrees.  So  also  may  the  angle  4  &  &•>  if  it  be  known, 
be  laid  off  frdm  the  scale,  inasmuch  as  the  lines  4  $  and  5  3 
appear  just  as  they  would  if  they  were  in  a  vertical  plane 
that  coincided  with  1  £•* 

*  Note. — The  scale  shown  in  Fig.  22  may  be  applied  to  the  drawing  of 
definite  angles  in  cabinet  perspective,  when  the  measures  of  angles  can  be 
taken  on  that  edge  of  a  cabinet  square  which  measures  the  same  number  of 
spaces  as  the  edge  of  a  corresponding  rectangular  square. 

Thus  in  the  cabinet  cube  7?,  Fig.  1,  page  1,  of  Drawing-Book  Xo.  II., 
which  is  a  cube  of  six  spaces  (six  inches)  to  a  side,  angles  at  5  or  3,  up  to 
forty -five  degrees,  maybe  taken  from  the  scale  and  laid  ofFon  the  side  6  4l 


ISOMETRICAL   DRAWING.  197 

IV.  THE  ISOMETRIC  ELLIPSE  AND  ITS  APPLICATIONS. 

The  Isometric  Ellipse  is  the  ellipse  which  is  drawn  within 
an  isometric  square,  touching  the  middle  points  of  its  sides, 
as  the  three  ellipses  in  Fig.  26,  Plate  V.  The  isometric 
ellipse  represents  a  circle  viewed  in  the  position  of  a  side 
of  an  isometric  cube.* 

Fig.  26.  Plate  V. — Here  is  represented  a  cube  which  meas- 
ures ten  spaces  to  a  side,  and  on  each  of  its  three  visible 
faces  is  an  isometric  ellipse  which  represents  a  circle  drawn 
touching  the  middle  of  the  sides  of  the  inclosing  square. 

and  angles  at  4  antl  &•>  UP  to  forty-five  degrees,  may  be  laid  off  on  the  side 
53.  So  angles  at  3  and  1,  up  to  forty-five  degrees,  may  be  laid  off  on  the 
side  24;  and  angles  at  8  and  ^,  up  to  forty-five  degrees,  may  be  laid  off  on 
the  side  1  3.  Angles  for  the  front  face  may  be  laid  off  on  all  the  sides  of 
that  face.  But  to  lay  off  an  angle  at  £,  on  the  line  3  4 — although  the  meas- 
ure 3  4  would  make  the  angle  3  6  4  one  of  forty-five  degrees,  yet  for  lesser 
angles  we  should  be  obliged  to  take  such  proportions  of  3  4  as  the  measures 
for  angles,  on  the  scale,  bear  to  the  entire  side  of  the  rectangular  square 
from  which  the  measures  are  taken.  It  would  be  the  same  when  an  angle 
at  4  °r  3  should  be  required  to  be  laid  off  on  the  side  12;  or  an  angle  at 
1  or  2  should  be  required  to  be  laid  off  on  the  side  3  4- 

The  same  principles  apply  to  the  laying  off  of  angles  in  semi-diagonal 
cabinet  perspective.  See  pages  10  and  11  of  Drawing-book  No.  IV.  Yet, 
for  practical  purposes  in  all  working  drawings,  the  true  angles,  or  inclina- 
tions of  lines,  can  generally  best  be  laid  off  by  some  known  measurements 
on  the  objects  themselves. 

*  Note. — In  the  isometric  ellipse,  what  is  called  the  major  axis  (greater 
diameter)  is  a  liule  more  than  once  and  seven  tenths  the  length  of  the  minor 
axis ;  and  it  is  of  the  same  length  as  the  diameter  of  the  circle  which  the 
ellipse  represents.  Thus,  in  Fig.  26,  the  upper  ellipse  represents  a  circle 
whose  diameter  is  s  t—  that  is,  it  represents  the  outer  circle  of  Fig.  27,  Plate 
VI.,  while  the  inner  circle  of  Fig.  27  is  the  one  we  are  obliged  to  compare 
it  with  in  prescribing  the  rules  of  practical  isometrical  drawing.  The  rea- 
son of  this  is  that  the  square  within  which  the  circle  is  drawn  is  dimin- 
ished in  apparent  length  of  sides  by  an  isometrical  view  of  it;  and  we 
adapt  the  scale  of  our  drawing  to  the  apparent  size,  and  not  to  the  real 
size.  Hence  we  draw  a  rectangular  square,  as  A  B  CD,  of  Fig.  27,  hav- 
ing the  same  real  length  of  sides  as  the  apparent  length  of  the  sides  of  the 
isometric  square,  A  B  C  D,  of  Fig.  26 ;  and  then  any  lines,  divisions,  or 
points  of  the  one  may  have  corresponding  lines,  divisions,  and  points  in  the 
other.  That  is,  both  may  be  drawn  to  the  same  s^ah ;  and  one  may  be 
used  to  illustrate  the  other.  Thus  the  two  kinds,  cabinet  and  isometrical 
drawing,  perfectly  harmonize  in  measurement. 


198  APPENDIX. 

Taking,  first,  the  upper  face  of  the  cube  for  illustration, 
we  see  that  it  is  an  isometrical  square  of  ten  spaces  (ten 
feet)  to  a  side,  and  crossed  by  equidistant  isometrical  lines 
parallel  to  the  sides.  In  Fig.  27,  Plate  VI.,  we  have  the  rect- 
angular square  A  I>  CD,  of  ten  spaces  (ten  feet)  to  a  side, 
and  also  crossed  by  the  same  number  of  equidistant  lines 
parallel  to  the  sides.  A  circle  is  also  drawn  within  this 
rectangular  square  touching  the  middle  points  of  its  four 
sides,  which  circle  is  represented  by  the  ellipse  of  Fig.  26. 
Now,  as  the  inner  circle  of  Fig.  27  is  a  circle  of  five  spaces' 
radius,  the  circumference  passes  through  the  twelve  num- 
bered points  of  the  intersections  of  the  ruled  lines,  as  there 
designated  from  1  to  12  inclusive.  (See  page  150.)  The  el- 
lipse of  Fig.  26  must  therefore  pass  through  the  same  num- 
ber of  corresponding  points  in  the  ruling,  so  that  we  thus 
have  twelve  definite  points  through  which  the  ellipse  must 
be  drawn.  The  ellipse  may  therefore,  by  these  aids,  be 
drawn  quite  accurately  by  the  hand  alone,  by  tracing  a 
symmetrical  curve  through  these  twelve  points.  The  same 
holds  good  as  to  the  ellipses  on  the  other  two  visible  faces 
of  the  cube. 

Any  isometric  ellipse  that  represents  a  circle  of  ten,  twen- 
ty, thirty,  forty,  etc.,  spaces'  diameter,  may  thus  have  twelve 
of  its  points  given.  But  when  the  ellipses  represent  circles 
of  other  proportions,  they  must  be  drawn  by  the  aid  of  the 
following  principles  and  methods : 

Scale  of  Diameters  and  Axes  of  Isometric  Ellipses. 
In  every  isometric  ellipse  there  is.  in  addition  to  the  ma- 
jor and  the  minor  axis,  what  is  called  the  isometric  diam- 
eter. Thus,  in  the  upper  ellipse  of  Fig.  26, 1  7  or  4  10  is  the 
isometric  diameter  of  the  ellipse — its  position  being  central- 
ly equidistant  from,  and  parallel  to,  the  sides  of  the  inclos- 
ing isometric  square.  The  isometric  diameter  is  equal  to  a 
side  of  the  isometric  square.  Hence,  when  an  isometric 
square  is  laid  down  on  isometrically  ruled  paper,  the  iso- 
metric diameter  of  the  ellipse  that  may  be  drawn  within  it 
is  also  known,  and  may  be  located  by  merely  counting  the 
spaces  on  either  of  the  side  lines  of  the  square. 


ISOMETEICAL  DRAWING.  199 

The  proportions  which  the  minor  axis,  the  major  axis,  and 
the  isometric  diameter  bear  to  one  another  are  also  known ; 
and  a  table  of  these  relative  proportions  is  given  on  page 
205.  We  have  also  prepared,  in  Fig.  28,  Plate  YL,  a  diagram 
scale,  in  which  the  proportions  are  given,  by  measure,  for  el- 
lipses of  any  size  up  to  one  whose  isometric  diameter  is  not 
more  than  thirty-five  spaces  of  the  isometric  ruling  given 
on  the  isometric  drawing-paper.  The  scale,  however,  may 
easily  be  extended  to  any  required  size. 

Fig.  28.  Plate  VI. — To  illustrate.  The  scale  is  made  in 
the  following  manner:  Take  a  rectangular  square  of  any 
equal  number  of  spaces  to  a  side,  as  A  B  CD.  From  one 
corner,  as  D,  with  a  radius  D  B,  cut  the  side  D  C  extend- 
ed, in  E,  and  join  E  A.  From  the  points  where  the  vertical 
ruled  lines  from  above  intersect  the  line  A  E,  draw  hori- 
zontal lines  to  the  line  A  D.  These  thirty-five  horizontal 
lines,  thus  drawn,  measuring  from  one  space  up  to  thirty- 
five  spaces,  represent  the  isometric  diameters  of  that  number 
of  ellipses,  while  to  each  isometric  diameter  is  assigned  its 
proper  major  axis  and  minor  axis. 

Thus,  if  the  isometric  diameter  of  the  ellipse  be  the  line 
D  E,  its  major  axis  will  be  A  E,  and  its  minor  axis  A  D. 
Again:  if  the  isometric  diameter  of  a  required  ellipse  be 
twenty  spaces,  its  measure  will  be  the  horizontal  line  from 
the  point  20,  on  the  diagonal  line  A  E,  to  the  line  AD; 
its  major  axis  will  be  the  measure  from  20  to  A;  and  its 
minor  axis  will  be  the  measure  from  the.  point  A,  down  to 
the  intersection  of  the  line  A  D,  with  the  horizontal  line 
drawn  from  20.  Or,  what  is  the  same  thing,  the  minor  axis 
will  be  the  measure  from  the  point  20  up  to  the  point  t  on 
the  line  A  B. 

Fig.  26.  Plate  V. — Application.  Suppose  that,  in  Fig. 
26,  we  have  merely  the  isometric  square  A  B  C  D,  of  ten 
spaces  to  a  side,  and  wish  to  draw  within  it  an  isometric  el- 
lipse touching  the  middle  points  of  its  four  sides.  The  iso- 
metric diameter  being  7  1  (or  its  equal,  B  C),  we  observe 
that  it  is  represented  by  the  horizontal  line  on  the  scale, 
Fig.  28,  from  10,  on  the  diagonal  line  A  E,  to  the  vertical 
line  A  D.  The  line  10  A  is,  then,  the  major  axis  of  the  el- 


200  APPENDIX. 

lipse,  and  10  j  the  minor  axis.  Therefore,  take  the  distance 
10  A  on  the  compasses,  and,  applying  it  to  Fig.  26,  lay  it 
off  on  the  line  13  J)  equidistant  on  both  sides  of  the  centre, 
c,  and  it  will  give  the  points  5  and  t,  the  extremes  of  the 
major  axis.  Also  take  the  distance  10  j  on  the  compasses, 
and,  applying  it  to  A  6r,  Fig.  26,  lay  it  off  equidistant  on 
both  sides  of  the  centre,  c,  and  it  will  give  the  points  u  and 
v,  the  extremes  of  the  minor  axis. 

In  this  same  manner  may  the  major  and  the  minor  axes 
of  the  ellipses  on  the  sides  of  the  cube  be  laid  off.  And  as 
the  diagram  scale  (Fig.  28)  may  be  easily  and  accurately 
made  of  any  required  size,  on  the  isometrical  drawing-pa- 
per, the  extreme  points  of  the  major  axis,  the  minor  axis, 
and  the  two  isometrical  diameters — eight  points  in  all — 
may  be  found  for  any  required  isometrical  ellipse.  Through 
these  points  the  curve  may  be  traced  by  hand  ;  or  it  may  be 
better  drawn  by  the  compasses,  with  great  approximate 
accuracy,  in  the  following  manner. 

To  Draw  the  Ellipse  by  the  Aid  of  the  Compasses. 

Fig.  29.  Let  A  IB  C  D  be  the  isometric  square  within 
which  the  ellipse  is  to  be  drawn.  Find  the  extreme  points, 
s  t  and  u  v9  of  the  major  and  the  minor  axis,  as  before 
shown.  Take  the  point  y  so  as  to  make  t  y  equal  to  t  D, 
and  from  y  describe  a  curve  passing  through  t  and  barely 
touching  the  sides  D  C  and  D  A.  The  other  end  curve  of 
the  ellipse  is  to  be  drawn  in  like  manner. 

Make  c  x  equal  to  c  C.  Take  the  distance,  t  x,  by  the 
compasses,  and  lay  it  off  from  A  to  z.  With  one  point  of 
the  compasses  in  z,  and  the  other  extended  to  1  or  10,  de- 
scribe the  side  curve  1  u  10.  It  should  pass  through  the 
extremity,  it,  of  the  minor  axis.  In  the  same  manner  find 
the  point  w  above  G",  and  from  it  describe  the  side  curve  4  v  7. 
In  this  manner  the  ellipse  will  be  so  accurately  drawn  that 
even  in  large  ellipses  the  eye  can  scarcely  detect  a  variation 
from  the  true  outline.* 

*  Note. — A  more  accurate  method  might  be  given  for  drawing  smrtll 
portions,  not  more  than  thirty  degrees  in  extent,  of  the  central,  side,  and  end 
curves  ;  and  the  points  for  describing  the  side  curves  would  be  a  \\tt\efar- 


ISCirilTBICAL   DEAWIXG.  201 

The  ellipse  of  Fig.  29  may  be  considered  the  npper  end 
of  a  vertical  cylinder,  having  an  axis,  c  #,  of  thirteen  feet, 
and  a  diameter,  1  7,  or  4  10,  of  ten  feet.  The  side  lines,  t  3 
and  s  5,  are  drawn  the  same  as  the  side  lines  of  vertical  cyl- 
inders in  cabinet  perspective,  while  the  ellipse  for  the  bot- 
tom— only  half  of  which  is  visible — is  drawn  within  the 
isometric  square,  E  F  G  H,  in  the  same  manner  as  the  up- 
per ellipse.  The  side  curves  on  the  cylinder  are  described 
from  points  below  w^  by  continued  removals  of  two  spaces 
downward,  and  all  with  the  same  stretch  of  the  compasses, 
w  v  or  w  4;  while  the  end  curves  are  described  in  a  similar 
manner,  from  points  vertically  below  y  and  n. 

Fig.  30  is  an  ellipse  representing  a  circle  of  only  five  spaces' 
(five  feet)  diameter,  described  on  the  top  of  a  vertically 
placed  block;  and  Fig.  31  represents  one  of  the  same  di- 
mensions on  the  end  of  a  horizontally  placed  block.  Fig. 
32  represents  a  cylinder,  six  feet  in  diameter  and  three  feet 
in  length,  placed  horizontally,  the  end  of  it  being  in  the 
same  position  as  the  ellipse  on  the  left-hand  side  of  Fig.  26. 
Fig.  33  is  a  cylinder  of  the  same  dimensions  as  Fig.  32,  but 
the  visible  end  of  it  is  in  the  position  of  the  ellipse  on  the 
right-hand  side  of  Fig.  26.  The  upper  part  of  Fig.  34  rep- 
resents a  vertical  cylinder,  two  feet  in  diameter  and  four  feet 
in  length,  cut  from  a  block  two  feet  square  at  the  end. 

PLATE  VII.— SCALE  OF  ONE  FOOT  TO  A  SPACE. 
Fig.  35  represents  a  hollow  cylinder,  fourteen  feet  in  ex- 
treme diameter,  four  feet  in  height,  and  having  its  wralls  one 
foot  in  thickness.  It  will  be  seen  that  its  upper  outer  ellipse 
is  drawn  within  the  isometric  square  A  J2  CD, and  that  the 
inner  ellipse  is  drawn  within  a  square  one  foot  within  the 
outer  square.  The  farther  inner  bottom  curve  "must,  evi- 
dently, be  drawn  within  a  square  of  the  same  dimensions  as 
the  upper  inner  square. 

ther  from  the  centre,  r,  than  those  we  have  given,  while  those  for  describii»g 
the  end  curves  would  be  a  trifle  nearer  the  centre,  c,  than  those  we  have 
given.  But  this  method  would  require  one  half  of  the  side  curves  of  the 
ellipse  to  be  drawn  without  the  aid  of  compasses ;  and  the  result  would  sel- 
dom be  as  accurate  as  by  the  method  we  have  given. 

12 


202  APPENDIX. 

Fig.  36  illustrates  a  method  of  dividing  the  ellipse  into 
any  number- of  equal  parts,  or  of  making  in  it  any  required 
divisions. 

From  E,  the  centre  of  a  side,  A  13,  of  the  isometrical 
square  which  incloses  the  ellipse,  draw  ED  at  right  angles 
to  A  It*  and  equal  to  E  A.  Connect  A  D  and  JB  D. 
From  .Z),  with  any  radius,  the  greater  the  better,  describe  a 
curve,  2  3,  cutting  the  lines  B  D  and  A  D.  Mark  this 
curve  according  to  the  divisions  required  in  A  c  D,  one 
quarter  of  the  ellipse,  and  through  the  points  of  division 
draw  lines  from  D  to  the  line  A  1>.  From  the  intersections 
of  these  lines  with  A  J3  draw  lines  to  the  centre,  c,  of  the 
ellipse,  and  the  quarter  part  of  the  ellipse  will  be  divided  in 
the  same  proportions  as  the  curve  2  3  is  divided.  Here 
the  curve  2  3  is  designed  to  be  divided  into  eight  equal 
parts — four  parts  on  each  side  of  the  central  point  E;  and 
hence  one  quarter  of  the  ellipse  is  divided  into  the  same  num- 
ber of  equal  parts.  If  the  same  divisions  are  to  be  continued 
throughout  the  ellipse,  transfer  the  points  of  division  on  A  J3 
to  the  other  sides  of  the  isometric  square,  and  from  them 
draw  lines  to  the  centre,  c,  etc. 

Or  the  method  given  in  Figs.  21,  22,  23,  and  24,  of 
Drawing-Book  No.  IV.,  may  be  adopted  for  all  isometrical 
ellipses. 


V.  MISCELLANEOUS  APPLICATIONS. 

Fig.  37.  To  draw  an  isometrical  octagon  within  an  iso- 
metrical square.  And,  1st,  when  one  of  the  sides  of  the  oc- 
tagon is  to  coincide,  in  part,  with  the  sides  of  the  isometrical 
square : 

Within  the  isometrical  square  lay  down  the  points  1  2 
and  3  4  (taken  from  diagram,  Fig.  28)  for  the  extremities 
of  the  major  and  the  minor  axis  of  the  ellipse  to  be  drawn 
within  the  square.  Then  through  these  points  draw  lines 
at  right  angles  to  the  two  axes,  and  the  lines  thus  drawn 

*  If  the  line  E  D  be  drawn  from  E  in  the  direction  of  two-space  diag- 
onals, it  will  be  at  right  angles  to  A  B. 


ISOilETKICAL   DRAWING.  203 

will  be  four  of  the  sides  of  the  required  octagon.  The  other 
four  sides  will  be  those  portions  of  the  sides  of  the  isometric 
square  lying  between  the  intersections  of  the  first  four  lines. 
It  will  be  seen  that  the  ruling  of  the  paper  is  a  perfect  guide 
for  drawing  lines  at  right  angles  to  the  major  and  the  mi- 
nor axis. 

2d.  When  the  centre  of  each  of  the  four  sides  of  the  iso- 
metric square  is  to  be  touched  by  an  octagonal  corner. 

Draw  two  lines  from  each  extremity  of  the  major  and 
minor  axes  to  the  centres  of  the  two  sides  adjacent  each  ex- 
tremity, and  the  octagon  will  be  completed.  Thus  draw 
lines  from  2  to  5  and  6,  from  4  to  5  and  £,  from  1  to  7  and 
8,  and  from  3  to  6  and  7.  Inner  lines  may  easily  be  drawn 
parallel  to  the  outer  lines. 

Fig.  38  is  an  octagonal  tub  or  box  nine  and  a  half  feet  in 
vertical  height ;  and  the  sides,  one  foot  in  thickness,  bevel 
outward  from  the  top  downward.  The  top  is  inclosed  by 
an  isometrical  square  of  ten  feet  to  a  side,  and  the  bottom 
by  a  square  of  thirteen  feet  to  a  side.  The  figure  itself  will 
sufficiently  explain  the  method  of  drawing  it. 

PLATE  VIII.— SCALE  OF  ONE  FOOT  TO  A  SPACE. 

Fig.  39  shows  that  the  half  of  a  cylinder,  of  three-feet  ra- 
dius, cut  longitudinally  and  vertically,  has  been  taken  from 
a  piece  of  timber  measuring  at  the  upper  end  four  feet  by 
six  feet.  The  hollow  in  the  timber  is  semicircular,  but 
shows  here  as  the  half  of  an  isometric  ellipse,  viewed  in  the 
position  of  the  ellipse  on  the  right-hand  side  of  the  cube  in 
Fig.  26. 

Fig.  40  represents  a  semicircular  arch  of  twelve-feet  span 
in  its  extreme  measure  from  2  to  3,  five  feet  in  length,  and 
with  walls  two  feet  in  thickness.  The  arch  shows  the  up- 
per halves  of  two  isometric  ellipses,  each  viewed  in  the  po- 
sition of  the  upper  half  of  the  ellipse  seen  on  the  right-hand 
side  of  the  cube  in  Fig.  26.  The  outer  ellipse  is  drawn 
within  an  isometrical  square  of  twelve  feet  to  a  side,  and 
the  inner  ellipse  within  one  of  eight  feet  to  a  side.  The 
farther  curve  is  part  of  an  ellipse  like  the  outer  front  ellipse. 
The  method  of  drawing  the  lines  for  the  uniform  layers  of 


204  APPENDIX. 

stones  that  form  the  arch  is  sufficiently  illustrated  by  the 
drawing  itself. 

Fig.  41  is  the  drawing  of  a  small  building,  and,  according 
to  the  scale  here  given,  it  is  only  six  by  eight  feet  on  the 
ground,  with  corner  posts  only  three  feet  high,  the  ridge 
rising  two  feet  above  the  level  of  the  tops  of  the  posts,  and 
the  chimney  two  feet  above  the  ridge.  Observe  how  the 
ridge  runs  centrally  over  the  building,  and  how  the  chimney 
is  placed  centrally  on  the  ridge,  and  also  equidistant  from 
the  two  extremes  of  the  ridge. 

Fig.  42  represents  a  structure  having  a  ground-plan  in 
the  form  of  a  cross.  The  four  roofs  have  sloping  ends  as 
well  as  sloping  sides,  and  are  what  are  called  hip-roofed. 
Moreover,  the  slope  of  the  ends  is  the  same  as  the  slope  of 
the  sides.  Thus  the  point  3  is  two  feet  above  the  level  of 
the  tops  of  the  posts ;  and  if  the  end  4  £,  and  the  side  4  5, 
were  extended  upward,  the  horizontal  distance  to  the  side 
would  be  the  same  as  the  horizontal  distance  to  the  end, 
being  three  feet  in  both  cases. 

Fig.  43  is  a  clustered  column,  formed  of  four  pieces,  each 
one  foot  square  at  the  upper  end,  but  each  beveling  outward 
below.  The  moulding  around  it  is  beveled  also,  to  corre- 
spond to  the  sides  of  the  column. 


With  the  aid  of  the  foregoing  illustrations  and  the  iso- 
metrical  drawing-paper,  the  student  ought  now  to  meet  with 
little  difficulty  in  applying  the  isometrical  method  of  repre- 
sentation to  all  objects  that  are  bounded  by  straight  lines 
or  by  regular  curves.  Irregular  surface  curves  may  also  be 
represented  isometrically  without  difficulty  by  first  drawing 
them  on  the  rectangular  ruled  paper,  from  which  they  may 
be  easily  transferred  to  the  isometrical  paper,  as  the  spaces 
on  both  measure  alike.  The  student  would  do  well  to  rep- 
resent, isometrically,  all  the  figures  and  problems  in  Draw- 
ing-Books II.,  III.,  and  IV. ;  and  he  will  generally  find  the 
change  quite  easy  from  the  cabinet  to  the  isometrical  draw- 
ing, if  he  understands  the  former. 


ISOMETKICAL   DRAWING. 


205 


TABLE  FOR  DRAWING  CIRCLES  Ds  ISOMETRICAL 
PERSPECTIVE. 

The  figures  in  the  columns  of  Isometrical  Diameters  denote  the  lengths 
of  isometric  diameters  (or  sides  of  isometric  squares) ;  and  the  figures  in 
the  other  two  columns  denote  the  corresponding  lengths  of  the  minor  and 
major  axes.  Thus,  if  an  ellipse  is  to  be  drawn  in  an  isometric  square  of  1 0 
spaces  to  a  side,  the  isometric  diameter  will  be  10  spaces  in  length,  the 
minor  axis  will  be  7.071  spaces  in  length,  and  the  major  axis  12.247  spaces 
in  length.  The  Table  gives  the  relative  proportions  of  the  isometric  diam- 
eters, minor  axes,  and  major  axes  for  all  isometric  ellipses  drawn  in  iso- 
metric squares  of  from  1  to  90  spaces  in  diameter.  The  principle  holds 
good  whatever  measure  of  length  the  figures  in  the  columns  of  Isometrical 
Diameters  represent. 


Ipom. 
Diam. 

Minor 
Axis. 

Major 
Axis. 

L*ora. 
Diam. 

Minor  1  Major 
Axis,  j  Axis. 

Isoin. 
Diam. 

Minor 
Axis. 

Major 
Axis. 

1 

.707 

1.225 

31 

21.920 

37.9G7 

61 

43.134 

74.709 

2 

1.414 

2.449 

32 

22.627 

39.192 

62 

43.841 

75.934 

3 

2.121 

3.674 

33 

23.335 

40.417 

63 

44.548 

77.159 

4 

2.828 

4.899 

34 

24.042 

41.641 

64 

45.255 

78.384 

5 

3.536 

6.124 

35 

24.749 

42.866 

65 

45.962 

79.608 

6 

4.243 

7.348 

36 

25.456 

44.091 

66 

46.669 

80.833 

7 

4.950 

8.573 

37 

26.163 

45.316 

67 

47.376 

82.058 

8 

5.657 

9.798 

38 

26.870 

46.540 

68 

48.083 

83.283 

9 

6.364 

11.023 

39 

27.577 

47.765 

69 

48.790 

84.507 

10 

7.071 

12.247 

40 

28.284 

48.990 

70 

49.497 

85.732 

11 

7.778 

13.472 

41 

28.991 

50.215 

71 

50.205 

86.957 

12 

8.485 

14.697 

42 

29.698 

51.439 

72 

50.912 

88.182 

13 

9.192 

15.922 

43 

30.406 

52.664 

73 

51.619 

89.406 

14 

9.899 

17.146 

44 

31.113 

53.889 

74 

52.326 

90.631 

15 

10.607 

18.371 

45 

31.820 

55.114 

75 

53.033 

91.856 

16 

11.314 

19.596 

46 

32.527 

56.338 

76 

53.740 

93.081 

17 

12.021 

20.821 

47 

33.234 

57.563 

77 

54.447 

94.300 

18 

12.728 

22.045 

48 

33.941 

58.783 

78 

55.154 

95.530 

19 

13.435 

23.270   49 

34.648 

60.012 

79 

55.861 

96.755 

20 

14.142 

24.495 

50 

35.355  61.237 

80 

56.569 

97.980 

21 

14.849 

25.720 

51 

36.062 

62.462 

81 

57.276 

99.204 

22 

15.556 

26.944 

52 

36.770 

63.687 

82 

57.983 

100.429 

23 

16.263 

28.169 

53 

37.477 

64.911 

83 

58.690 

101.654 

24 

16.971 

29.394 

54 

38.184 

66.136 

84 

59.397 

102.879 

25 

17.678 

30.619 

55 

38.891 

67.361 

85 

60.104 

104.103 

26 

18.385 

31.843 

56 

39.598 

68.586 

86 

60.811 

105.328 

27 

19.092 

33.068 

57 

40.305 

69.810 

87 

61.518 

106.553 

28 

19.799 

34.293 

58 

41.012 

71.035 

88 

62.225 

107.778 

29 

20.506 

35.518 

59 

41.719 

72.260 

89 

62.933 

109.002 

30 

21.213 

36.742 

GO 

42.426 

73.485 

90 

63.640 

110.227 

TTIIVEKSITY 


Scale  of  two  inches  to  a  space. 


PI.  I. 


Scale  of  one  foot  to  a  space. 


PI.  II. 


UNIVERSITY 


Scale  of  two  feet  to  a  space. 


PI.  in. 


Scale  of  one  foot  to  a  space. 


PI.  IV. 


J80 


150 


7 


I'igj.  2 


•Vr.    /V 


FT 


14 


1! 


is 


2ta 


UHIVERSIT7 


Scale  of  one  foot  to  a  space. 


PJ.  V 


Scale  of  one  foot  to  a  space. 


PL  VI. 


\ 


-^T 


s 


* 


\/ 


-/ 


\, 


T/ 

7 


K 


Scale  of  one  foot  to  a  space. 


PI.  VII. 


OP  THE 

•UHIVEKSIT7 


Scale  of  one  foot  to  a  space. 


PI.  VITT. 


UNIVEESITY  OF  CALIFOENIA  LIBEAEY, 
BEEKELEY 


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